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A HISTORY OF
THE CONCEPTIONS OF
LIMITS AND FLUXIONS
IN GREAT BRITAIN

FROM

NEWTON TO WOODHOUSE

[Frontispiece.

The Open Court Series of Classics of Science and
Philosophy, 3^o. 5

A HISTORY OF

THE CONCEPTIONS OF
LIMITS AND FLUXIONS
IN GREAT BRITAIN

FROM

NEWTON TO WOODHOUSE

FLORIAN CAJORI, PH.D.

PROFESSOR OF HISTORY OF MATHEMATICS IN THE
UNIVERSITY OF CALIFORNIA

With portraits of Berkeley and Maclaurin

CHICAGO AND LONDON

THE OPEN COURT PUBLISHING COMPANY

1919

303

TABLE OF CONTENTS

PAGE

INTRODUCTION i

CHAPTER I
NEWTON

Newton's Publications printed before 1734 2

Prineipia . ~ •' 3

Wallis's De Algebra Tractatus 14

Quadrature* Curvarum (1704) . . . . . 17

An Account of the Commercium Epistolicum . , .26

Newton's Correspondence and Manuscripts not in print in 1734 . 29

Remarks . " . . . 32

CHAPTER II
PRINTED BOOKS AND ARTICLES ON FLUXIONS

BEFORE 1734

John Craig, De Moivre, David Gregory, Fatio de Duillier, Cotes,

Ditton, Cheyne 37

John Harris, 1702, 1705, 1710 .40

Charles Hayes, 1704 .41

William Jones, 1706 ... . . . . . -43

Humphry Ditton, 1706 .43

Commercium Epistolicum D. Johannis Collins, 1712 . . . 47

Joseph Raphson, 1715 49

Brook Taylor, 1715 "... 50

James Stirling, 1717, 1730 50

Edmund Stone, 1730 . . ... . . . 50

Remarks . . ' . . -55

vi LIMITS AND FLUXIONS

CHAPTER III

BERKELEY'S ANALYST (1734); CONTROVERSY WITH
JURIN AND WALTON

PACK

The Analyst 57

Jurin's first reply to Berkeley 64

Walton's first reply to Berkeley 69

Berkeley's reply to Jurin and Walton ...... 72

Walton's second reply to Berkeley 78

Jurin's second reply to Berkeley 80

Berkeley's second reply to Walton ...... 85

The second edition of Walton's second reply .... 87

Remarks -89

CHAPTER IV

JURIN'S CONTROVERSY WITH ROBINS AND
PEMBERTON

Robins's Discourse on Fhixions ....... 96

Jurin's review of his own letters to Berkeley .... 101

Robins's rejoinder , 106

The debate continued . 109

Pemberton enters the debate . . . . . . .129

Debate over Robins's review of treatises by Leonhard Euler,

Robert Smith, and James Jurin . . . . . .139

Remarks . . . . . . . . . . .146

CHAPTER V

TEXT-BOOKS IMMEDIATELY FOLLOWING
BERKELEY'S ATTACK

John Colson, 1736 149

James Hodgson, 1736 . . . . . . . .155

Thomas Bayes, 1736 157

John Muller, 1736 ......... 162

Anonymous translation of Newton's Method of Fluxions, 1737 . 164

James Smith, 1737 165

Thomas Simpson, 1737 ........ 169

Benjamin Martin, 1739, 1759 . 171

An anonymous text, 1741 ........ 172

John Rowe, 1741, 1757, 1767 175

Berkeley ten years after . . 178

Remarks . .......... 179

TABLE OF CONTENTS vii

CHAPTER VI
MACLAURIN'S "TREATISE OF FLUXIONS, 1742"

PAGE

Remarks . 188

CHAPTER VII
TEXT-BOOKS OF THE MIDDLE OF THE CENTURY

John Stewart, 1745 190

William Emerson, 1743 (?), 1757, 1768 *92

Thomas Simpson, 1750 194

Nicholas Saunderson, 1756 ....... 197

John Rowning, 1756 198

Israel Lyons, 1758 201

William West, 1762 202

James Wilson, 1761 . . . 202

Remarks 206

CHAPTER VIII

ROBERT HEATH AND FRIENDS OF EMERSON IN CONTROVERSY
WITH JOHN TURNER AND FRIENDS OF SIMPSON

Robert Heath 207

Main articles in the controversy ....... 209

Ladies' Diary , 1751, 1752 ........ 219

Popular impression of the nature of fluxions .... 222

Remarks ........... 223

CHAPTER IX
ABORTIVE ATTEMPTS AT ARITHMETISATION

John Kirkby, 1748 225

John Petvin, 1750 230

John Landen, 1758 231

James Glenie, 1793 ... .... 235

Remarks ........... 238

viii LIMITS AND FLUXIONS

CHAPTER X
LATER BOOKS AND ARTICLES ON FLUXIONS

PAGE

Encyclopedia Britannica, 1771. 1779, 1797 .... 240

Robert Thorp, 1777 241

F. Holliday, 1777 243

Charles Hutton, 1796, 1798 244

S. Vince, 1795, l8°5 • 245

Agnesi — Colson — Hellins, 1801 ....... 247

T. Newton, 1805 250

William Dealtry, 1 810, 1816 252

New editions .......... 253

Remarks ........... 253

CHAPTER XI

CRITICISMS OF FLUXIONS BY BRITISH WRITERS UNDER THE
INFLUENCE OF D'ALEMBERT, LAGRANGE, • AND LACROIX

Review of Lagrange's Fonctions analytiques, 1 799 . . . 255

Review of a memoir of Stockier, 1799 . . 259

Review of Lacroix's Calcul differentiel, 1800 .... 260

Review of Czrnot's fitfexwrts, 1801 262

Robert Woodhouse, 1803 . . 263

William Hales, 1804 267

Encyclopedia Britannica^ 1810 . . . . . . . 269

Lacroix's Elementary Treatise, 1816. . . . . .271

Remarks ........... 274

CHAPTER XII

MERITS AND DEFECTS OF THE EIGHTEENTH-CENTURY
BRITISH FLUXIONAL CONCEPTIONS

Merits 277

Defects . . 279

Addenda 289

INDEX 294

LIMITS AND FLUXIONS

INTRODUCTION

I.' EVERY great epoch in the progress of science is
preceded by a period of preparation and prevision.
The invention of the differential and integral calculus
is said to mark a "crisis" in the history of mathe-
matics. The conceptions brought into action at
that great time had been long in preparation. The
fluxional idea occurs among the schoolmen — among
Galileo, Roberval, Napier, Barrow, and others. The
differences or differentials of Leibniz are found in
crude form among Cavalieri, Barrow, and others.
The undeveloped notion of limits is contained inV
the ancient method of exhaustion ; limits are found
in the writings of Gregory St. Vincent and many
others. The history of the conceptions which led
up to the invention of the calculus is so extensive
that a good-sized volume could be written thereon.
We shall not yield to the temptation of lingering
on this pre-history at this time, but shall proceed
at once to the subject-matter of the present
monograph.

CHAPTER I
NEWTON

2. IT was in the year 1734 that Bishop Berkeley
made his famous attack upon the doctrine of
fluxions, which was the starting-point of all philo-
sophical discussion of the new mathematics in
England during the eighteenth century. In what
follows we quote from the writings of Newton that
were printed before 1734 such parts as bear on his
conceptions of fluxions, so that the reader may
judge for himself what grounds there were for
Berkeley's great assault. To assist us in the inter-
pretation of some of these printed passages, we
quote also from manuscripts and letters of Newton
which at that time were still unprinted. In the
next chapter we give an account of the foundations
of fluxions as displayed by other writers in books
and articles printed in Great Britain before 1734.
It is hoped that the material contained in these first
two chapters will enable the student to follow closely
and critically the debates on fluxions.

From Newton's Publications printed before 1734

I. PRINCIPIA

3. Three editions of the Piincipia were brought
out in Newton's lifetime ; the first in 1687, the

NEWTON 3

second in 1713, the third in 1726. We give extracts
which bear on the theory of limits and fluxions and
indicate the changes in phraseology introduced in
the second and third editions. We give also trans-
lations into English based on the text of the 1726,
or third, edition.

Principia^ Book 7, Section 7, Lemma I
First edition :

4. " Quantitates, ut & quantitatum rationes, quae
ad aequalitatem dato tempore constanter tendunt
& eo pacto propius ad invicem accedere possunt
quam pro data quavis differentia ; fiunt ultimo
aequales.

5. "Si negas, sit earum ultima differentia D.
Ergo nequeunt propius ad sequalitatem accedere
quam pro data differentia D : contra hypothesin."

Second and third editions :

6. ' ' Quantitates, ut & quantitatum rationes, quae
ad aequalitatem tempore quovis finito constanter
tendunt, & ante finem temporis illius propius ad
invicem accedunt quam pro data quavis differentia,
fiunt ultimo aequales.

7. "Si negas, fiant ultimo inaequales, & sit
earum, etc." [As in the first edition.]

Translation by Robert Thorp:1

8. ' ' Quantities, and the ratios of quantities, which,
in any finite time, tend continually to equality; and

1 Mathematical Principles of Natural Philosophy, by Sir Isaac
Newton, Knight. Translated into English, and illustrated with a
Commentary, by ROBERT THORP, M.A., vol. i, London, 1777.

LIMITS AND FLUXIONS

before the end of that time, approach nearer to each
other than by any given difference, become ulti-
mately equal.

" If you deny it, let them be ultimately unequal ;
and let their ultimate difference be D. Therefore,
they cannot approach nearer to equality than by
that given difference D. Which is against the
supposition."

Principia, Book /, Section /, Lemma II
Translation by Motte : x

9. "If in any figure AaclL, terminated by the
right lines Aa, AE, and the curve acE, there be
inscribed any number of paral-
lelograms Ab, B<r, Co7, etc.,
comprehended under equal bases
AB, BC, CD, etc., and the sides
B£, O, D<^, etc., parallel to one
side Aa of the figure ; and the
parallelograms aKbl, b^Lcm,
cMdn, etc., are completed.
Then if the breadth of those
parallelograms be supposed to

be diminished, and their number be augmented in
infinitum ; I say, that the ultimate ratios which
the inscribed figure AK^LdVI^D, the circumscribed
figure AalbincndolL, and curvilinear figure AabcdE,
will have to one another, are ratios of equality.

1 The Mathematical Principles of Natural Philosophy, by Sir Isaac
Newton; translated into English dy ANDREW MOTTE, London, 1729.
(Two volumes.)

BF C D

FIG. i.

NE WTON 5

"For the difference of the inscribed and circum-
scribed figures is the sum of the parallelograms K/,
L;«, MX Do, that is (from the equality of all their
bases), the rectangle under one of their bases K£
and the sum of their altitudes Aa, that is, the
rectangle AB/#. But this rectangle, because its
breadth AB is supposed diminished in infinitum,
becomes less than any given space. And therefore
(by Lem. I) the figures inscribed and circumscribed
become ultimately equal one to the other ; and
much more will the intermediate curvilinear figure
be ultimately equal to either. O. E. D. "

Principia, Book /, Section /, Lemma XI,
Scholium (first part omitted)

10. ". . . Quae de curvis lineis deque super-
ficiebus comprehensis demonstrata sunt, facile appli-
cantur ad solidorum superficies curvas & contenta.
Praemisi vero haec Lemmata, ut effugerem taedium
deducendi perplexas1 demonstrationes, more veterum
Geometrarum, ad absurdum. Contractiores enim
redduntur demonstrationes per methodum Indivisi-
bilium. Sed quoniam durior est Indivisibilium
hypothesis, & propterea methodus ilia minus Geo-
metrica censetur ; malui demonstrationes rerum
sequentium ad ultimas quantitatum evanescentium
summas & rationes, primasque nascentium, id est,
ad limites summarum & rationum deducere ; &
propterea limitum illorum demonstrationes qua potui
brevitate praemittere. His enim idem prsestatur

1 In the third edition " longas" takes the place of " perplexas."

6 LIMITS AND FLUXIONS

quod per methodum Indivisibilium ; & principiis
demonstrates jam tutius utemur. Proinde in sequen-
tibus, siquando quantitates tanquam ex particulis
constantes consideravero, vel si pro rectis usurpavero
lineolas curvas ; nolim indivisibilia, sed evanescentia
divisibilia, non summas & rationes partium deter-
minatarum, sed summarum & rationum limites
semper intelligi ; vimque talium demonstrationum
ad methodum praecedentium Lemmatum semper
revocari.

II. "Objectio est, quod quantitatum evanes-
centium nulla sit ultima proportio ; quippe quae,
antequam evanuerunt, non est ultima, ubi evanu-
erunt, nulla est. Sed & eodem argumento aeque
contendi posset nullam esse corporis ad certum
locum pergentis x velocitatem ultimam. Hanc enim,
antequam corpus attingit locum, non esse ultimam,
ubi attingit, nullam esse. Et responsio facilis est.
Per velocitatem ultimam intelligieam,2 qua corpus
movetur ; neque antequam attingit locum ultimum
& motus cessat, neque postea, sed tune cum
attingit ; id est, illam ipsam velocitatem quacum
corpus attingit locum ultimum & quacum motus
cessat. Et similiter per ultimam rationem quanti-
tatum evanescentium intelligendam esse rationem
quantitatum non antequam evanescunt, non postea,
sed quacum evanescunt. Pariter & ratio prima

1 In the second and third editions "pervenientis" takes the place of
"pergentis." In the third edition the sentence reads, "... ad
certum locum, ubi motus finiatur, pervenientis velocitatem ultimam."

2 In the second and third editions "intelligi earn" takes the place
of "intelligieam."

NE WTON 7

nascentium est ratio quacum nascuntur. Et summa
prima & ultima est quacum esse (vel augeri et 1
minui) incipiunt & cessant. Extat limes quern
velocitas in fine motus attingere potest, non autem
transgredi. Haec est velocitas ultima Et par est
ratio limitis quantitatum & proportionum omnium
incipientium & cessantium. Cumque hie limes sit
certus & definitus, problema est vere Geometricum
eundem determinare. Geometrica vero omnia in
aliis geometricis determinandis ac demonstrandis
legitime usurpantur.

12. " Contendi etiam potest, quod si dentur
ultimae quantitatum evanescentium rationes, dabun-
tur & ultimae magnitudines ; & sic quantitas omnis
constabit ex indivisibilibus, contra quam Euclides
de incommensurabilibus, in libro decimo Elemen-
torum, demonstravit. Verum haec objectio falsae
innititur hypothesi. Ultimae rationes illae quibuscum
quantitates evanescunt, revera non sunt rationes
quantitatum ultimarum, sed limites ad quos quanti-
tatum sine limite decrescentium rationes semper
appropinquant, & quas propius assequi possunt quam
pro data quavis differentia, nunquam vero trans-
gredi, neque prius attingere quam quantitates
diminuuntur in infmitum. Res clarius intelligetur
in infinite magnis. Si quantitates duae quarum data
est differentia augeantur in infinitum, dabitur harum
ultima ratio, nimirum ratio aequalitates nee tamen
ideo dabuntur quantitates ultimae seu maximae
quarum ista est ratio. Igitur in sequentibus,

1 In the third edition " aut " takes the place of " et."

8 LIMITS AND FLUXIONS

siquando facili rerum imagination! l consulens,
dixero quantitates quam minimas, vel evanescentes
vel ultimas ; cave intelligas quantitates magnitudine
determinatas, sed cogita semper diminuendas sine
limite.",

Translation by Robert Thorp :

13. "Those things which have been demonstrated
of curve lines, and the surfaces which they compre-
hend, are easily applied to the curve surfaces and
contents of solids. But I premised these lemmas
to avoid the tediousness of deducing long demon-
strations to an absurdity, according to the method
of the ancient geometers. For demonstrations are
rendered more concise by the method of indivisibles.
But, because the hypothesis of indivisibles is some-
what harsh, and therefore that method is esteemed
less geometrical, I chose rather to reduce the
demonstrations of the following propositions to the
prime and ultimate sums and ratios of nascent and
evanescent quantities ; that is, to the limits of those
sums and ratios : and so to premise the demonstra-
tions of those limits, as briefly as I could. For
hereby the same thing is performed, as by the
method of indivisibles ; and those principles being
demonstrated, we may now use them with more
safety. Therefore, if hereafter I shall happen to
consider quantities, as made up of particles, or shall
use little curve lines for right ones, I would not be
understood to mean indivisible, but evanescent

1 In the third edition " conceptui>" takes the place of 'imagination!. "

NE WTON 9

divisible quantities ; not the sums and ratios of
determinate parts, but always the limits of sums
and ratios : and, that the force of such demonstra-
tions always depends on the method laid down in
the preceding lemmas.

14. "It is objected, that there is no ultimate
proportion of evanescent quantities ; because the
proportion, before the quantities have vanished, is
not ultimate ; and, when they have vanished, is
none. But, by the same argument, it might as
well be maintained, that there is no ultimate
velocity of a body arriving at a certain place, when
its motion is ended : because the velocity, before
the body arrives at the place, is not its ultimate
velocity ; when it has arrived, is none. But the
answer is easy : for by the ultimate velocity is
meant that, with which the body is moved, neither
before it arrives at its last place, when the motion
ceases, nor after ; but at the very instant when it
arrives ; that is, that very velocity with which the
body arrives at its last place, when the motion
ceases. And, in like manner, by the ultimate ratio
of evanescent quantities is to be understood the
ratio of the quantities, not before they vanish, nor
after, but that with which they vanish. In like
manner, the first ratio of nascent quantities is that
with which they begin to be : and the first or last
sum is that, with which they begin and cease to be,
or to be augmented or diminished. There is a
limit, which the velocity at the end of the motion
may attain, but cannot exceed. This is the

io LIMITS AND FLUXIONS

ultimate velocity. And there is a like limit in all
quantities and proportions that begin and cease to
be. And since such limits are certain and definite,
to determine the same is a problem strictly geo-
metrical. But whatever is geometrical we may be
allowed to use in determining and demonstrating
any other thing that is likewise geometrical.

15. " It may be also argued, that if the ultimate
ratios of evanescent quantities are given, their
ultimate magnitudes will be also given ; and so all
quantities will consist of indivisibles, which is con-
trary to what Euclid has demonstrated concern-
ing incommensurables, in the tenth book of his
Elements. But this objection is founded on a false
supposition, for those ultimate ratios with which
quantities vanish are not truly the ratios of ultimate
quantities, but the limits to which the ratios of
quantities, decreasing without end, always con-
verge ; and to which they may approach nearer
than by any difference, but can never go beyond,
nor attain to, unless the quantities are diminished
indefinitely. This will appear more evident in
quantities indefinitely great. If two quantities,
whose difference is given, are augmented continu-
ally, their ultimate ratio will be given, to wit, the
ratio of equality ; but the ultimate or greatest
quantities themselves, of which that is the ratio,
will not therefore be given. If then in what follows,
for the more easy apprehension of things, I shall
ever mention quantities the least possible, or evanes-
cent^ or ultimate^ beware lest you understand quan-

NE WTON 1 1

tities of a determinate magnitude ; but conceive
them to be continually diminished without limit."

Principia, Book 77, Section 77, Lemma 77.

1 6. " . . . Has quantitates, ut indeterminatas
& instabiles, & quasi motu fluxuve perpetuo cres-
centes vel decrescentes, hie considero ; & earum l
incrementa vel decrementa momentanea sub nomine
momentorum intelligo : ita ut incrementa pro
momentis addititiis seu affirmativis, ac decrementa
pro subductitiis seu negativis habeantur. Cave
tamen intellexeris particulas fmitas. Momenta,
quam primum finitae sunt magnitudinis, desinunt
esse momenta. Finiri enim repugnat aliquatenus
perpetuo eorum incremento vel decremento.2 In-
telligenda sunt principia jamjam nascentia fmitarum
magnitudinum. Neque enim spectatur in hoc
lemmate magnitude momentorum, sed prima nas-
centium proportio. Eodem recidit si loco momen-
torum usurpentur vel velocitates incrementorum ac
decrementorum (quas etiam motus, mutationes &
fluxiones quantitatum nominare licet) vel finitae
quaevis quantitates velocitatibus hisce proportion-
ales. Termini3 autem cujusque generantis coeffi-
ciens est quantitas, quse oritur applicando genitam

1 The first edition gives " eorum " instead of " earum."

2 In the place of" Momenta, quam primum finitae sunt magnitudinis,
desinunt esse momenta. Finiri enim repugnat aliquatenus perpetuo
eorum incremento vel decremento," the second and third editions have
this: " Particulae finitae non sunt momenta, sed quantitates ipsae ex
momentis genitse."

3 In the second and third editions " Lateris " takes the place of
" Termini."

12 LIMITS AND FLUXIONS

ad hunc terminum. Igitur sensus lemmatis est,
ut, si quantitatum quarumcunque perpe^tuo motu
crescentium vel decrescentium A, B, C, etc.,
momenta, vel mutationum velocitates 2 dicantur
a, b, c, etc., momentum vel mutatio geniti 3 rectan-
guli AB fuerit «B+£A, et geniti3 contenti ABC
momentum fuerit «BC +£AC+<:AB : . . .

17. " Cas. i. Rectangulum quodvis motu perpetuo
auctum AB, ubi de lateribus A & B deerant momen-
torum dimidia4 \a & \by fuit A — \a in B — \b,
seu AB — \ <zB — -J- bA + J ab ; & quam primum latera
A & B alteris momentorum dimidiis aucta sunt,
evadit A + J0inB + J b, seu AB + \ aft + \ b A + J ab.
De hoc rectangulo subducatur rectangulum prius, et
manebit excessus #B + <£A. Igitur laterum incre-
mentis totis a et b generatur rectanguli incrementum
Q.E.D."

English Translation by Andrew Motte :

1 8. '* . . . These quantities I here consider as
variable and indetermined and increasing or decreas-
ing as it were by perpetual motion or flux ; and I
understand their momentaneous increments or
decrements by the name of Moments ; so that the

1 In the second and third editions "hoc latus " takes the place of
"hunc terminum."

- In the third edition " vel his proportionales mutationum veloci-
tates" takes the places of "vel mutationum velocitates."

3 "geniti" is left out in the first edition.

4 In this history, the solidus ( / ) will be used sometimes in printing
fractions which come in the line of the text. The reader must remem-
ber that this notation is modern ; it occurs in none of the passages
which we quote from seventeenth- and eighteenth-century books. In
some cases the use of the solidus has made it necessary to insert paren-
theses which do not occur in the original.

NEWTON 13

increments may be esteemed as added, or affirm-
ative moments ; and the decrements as subducted,
or negative ones. But take care not to look upon
finite particles as such. Finite particles are not
moments, but the very quantities generated by the
moments. We are to conceive them as the just
nascent principles of finite magnitudes. Nor do we
in this Lemma regard the magnitude of the moments,
but their first proportion as nascent. It will be the
same thing, if, instead of moments, we use either
the Velocities of the increments and decrements
(which may also be called the motions, mutations,
and fluxions of quantities) or any finite quantities
proportional to those velocities. The coefficient of
any generating side is the quantity which arises by
applying the Genitum to that side. Wherefore the
sense of the Lemma is, that if the moments of any
quantities A, B, C, etc., increasing or decreasing by
a perpetual flux, or the velocities of the mutations
which are proportional to them, be called a, b, c, etc. ,
the moment or mutation of the generated rectangle
AB will be ^B + ^A ; the moment of the generated
content ABC will be a^C + b AC + <:AB : . . .

19. " Case i. Any rectangle as AB augmented by
a perpetual flux, when, as yet, there wanted of the
sides A and B half their moments \a and-J^, was
A — \ a into B — } b, or AB — | # B — £ £A + J ab ; but
as soon as the sides A and B are augmented by the
other half moments ; the rectangle becomes A + \ a
into B + J b, or AB + \ a¥> + \ £A + \ ab. From this
rectangle subduct the former rectangle, and there

14 LIMITS AND FLUXIONS

will remain the excess #B + £A. Therefore with the
whole increments a and b of the sides, the increment
aE + £A of the rectangle is generated. Q. E. D. "

II. WALLIS'S DE ALGEBRA TRACTATUS

20. The Latin edition of John Wallis's Algebra,
which appeared in 1693, contains on pages 390-396
a treatise on the " Quadrature of Curves" which
Newton had prepared many years before, and from
which he cited many things in his letter of October
24, 1676. In revised phraseology and with a new
Introduction, the "Quadrature of Curves" was
republished in 1704, as we shall see presently.
Through the researches of Rigaud 1 we know now
that what is given in Wallis's Algebra, p. 390, line
1 8, to p. 396, line 19, are Newton's own words,
except, no doubt, the word "clarissimus," as
applied to himself. From this part we quote as
follows : 2 —

21. Page 391 : "Per fluent es quantitates intelli-
git indeterminatas, id est quae in generatione
Cuvarum per motum localem perpetuo augentur vel
diminuuntur, & per earum fluxionem intelligit celeri-
tatem increment! vel decrementi. Nam quamvis

fluentes quantitates & earum ftuxiones prima fronte
conceplfu difficiles videantur, (solent enim nova
difficilius concipi), earundem tamen notionem cito
faciliorem evasuram putat, quam sit notio momen-

1 S. P. Rigaud, Historical Essay on Sir Isaac Newton's Principia^
Oxford, 1838, p. 22.

2 Johannis Wallis, S.T.D., De Algebra Tractatus ; Historicus fr
Practicus. Oxoniae, MDCXCIII.

NEWTON 15

torum aut partium minimarum vel differentiarum
infinite paru arum : propterea quod figurarum & quan-
titatum generatio per motum continuum magis
naturalis est & facilius concipitur, & Schemata in
hac methodo sclent esse simpliciora, quam in ilia
partium. ..."

22. Page 392: " Sint v, x, y, z fluentes quanti-
tates, & earum fluxiones his notis vt x> j>, z desig-
nabuntur respective. Et quoniam hae fluxiones sunt
etiam indeterminatae quantitates, & perpetua muta-
tione redduntur majores vel minores, considerat
velocitates quibus augentur vel diminuuntur tanquam
earum fluxiones, & punctis binis notat in hunc
modum z), x, y, z, & perpetuum incrementum vel
decrementum harum fluxionum considerat ut ipsarum
fluxiones, ..."

23. Page 392: "Sit enim o quantitas infinite
parva, & sint oz, oy, ox Synchrona momenta seu
incrementa momentanea quantitatum fluentium *, 7,
& x : & hae quantitates proximo temporis momento
per accessum incrementorum momentaneorum eva-
dent z + oz, y + oy, x+ox\ ..." After substitut-
ing these in x* — xyy-^-aaz — Q, then subtracting the
original expression and dividing the remainder by
o, he remarks (page 393) : " Terminos multiplicatos
per o tanquam infinite parvos dele, & manebit
aequat i o $x& — xyy — 2xyy + aaz = o. "

Translation :

24. Page 391: "By flowing quantities he under-
stands indeterminates, that is, those which, in the

1 6 LIMITS AND FLUXIONS

generation of curves by local motion are always
increased or diminished, and by their fluxions he
understands the velocity of increase or decrease.
For, however difficult of comprehension flowing
quantities and their fluxions appear at first sight
(for new things are usually perceived with diffi-
culty), yet he thinks a notion of them will be
obtained more easily than the notion of moments
either of least parts or of infinitely small differences ;
because the generation of figures and quantities is
more naturally and easily conceived, and the draw-
ings in this method are usually more simple than in
that of parts."

25. Page 392: "Let the flowing quantities be
designated v, x, y, z, and their fluxions by the
marks ?>, x> y, s, respectively. And since these
fluxions are likewise indeterminate quantities, and
by perpetual motion become greater or lesser, he
considers the velocities by which they are increased
or diminished as their fluxions, and marks them
with double dots in this way v, x, y, z, and he con-
siders the perpetual increase or decrease of these
fluxions as fluxions of themselves. ..."

26. Page 392: "Let o be an infinitely small
quantity, and oz, oy, ox the synchronous moments
or momentaneous increments of the flowing quanti-
ties z, y, x : and these quantities at the next
moment of time, by the accession of the momen-
taneous increments become z + oz, y + oy, x + ox:
. . ." After substituting these in x*— xyy + aaz = o,
then subtracting the original expression and divid-

NEWTON 17

ing the remainder by o, he remarks (page 393) :
"Destroy the terms multiplied by o as infinitely
small, and there will remain the equation

III. QUADRATURA CURVARUM,1 1 704
" INTRODUCTIO

27. " Quantitates Mathematicas non ut ex parti-
bus quam minimis constantes, sed ut motu con-
tinuo descriptas hie considero. Lineae describuntur
ac describendo generantur non per appositionem
partium sed per motum continuum punctorum,
superficies per motum linearum, solida per motum
superficierum, anguli per rotationem laterum, tem-
pora per fluxum continuum, et sic in caeteris. Hae
Geneses in rerum natura locum vere habent et in
motu corporum quotidie cernuntur. Et ad hunc
modum Veteres ducendo rectas mobiles in longi-
tudinem rectarum immobilium genesin docuerunt
rectangulorum.

28. "Considerando igitur quod quantitates sequa-
libus temporibus crescentes et crescendo genitae,
pro velocitate majori vel minori qua crescunt ac

1 Tractatus de Quadratura Curvarutn, published in 1 704 in London,
as an appendix to Newton's Opticks. It was reprinted under the
editorship of William Jones in London in the year 1711, in a volume
containing also three other papers of Newton, viz., the De analyst per
aqnationes infinitas, Enumeratio linearum tertii ordinis, and Methodus
differentialis. An English translation of the Quadratura Curvarum,
made by John Stewart, was brought out in 1745 at London, in a volume
containing also Newton's Analysis by Equations of an Infinite Number
of Terms. A German translation of the Quadratura Curvarum by
Gerhard Kowalewski appeared at Leipzig in 1908 in Ostw aid's Klassiker
der exakten Wis sense haft en, Nr. 164.

LIMITS AND FLUXIONS

generantur, evadunt majores vel minores ; metho-
dum quaerebam determinandi quantitates ex veloci-
tatibus motuum vel incrementorum quibus gener-
antur; et has motuum vel incrementorum velocitates
nominando Fluxiones et quantitates genitas nomin-
ando Fluentes, incidi paulatim Annis 1665 et 1666
in Methodum Fluxionum qua hie usus sum in
Quadratura Curvarum.

29. " Fluxiones sunt quam proxime ut Fluentium

augmenta aequalibus temporis particulis quam mini-
mis genita, et ut accurate loquar, sunt in prima
ratione augmentorum nascentium ; exponi autem
possunt .per lineas quascunque quae sunt ipsis pro-
portionales. Ut si arese ABC, ABDG ordinatis
BC, BD super basi AB uniformi cum motu pro-
gredientibus describantur, harum arearum fluxiones
erunt inter se ut ordinatae describentes BC et BD,
et per ordinatas illas exponi possunt, propterea quod
ordinatae illae sunt ut arearum augmenta nascentia.

NEWTON 19

Progrediatur ordinata BC de loco suo BC in locum
quemvis novum be. Compleatur parallelogrammum
BCE£, ac ducatur recta VTH quae curvam tangat
in C ibsisque be et BA productis occurrat in T et
V : et abscissae AB, ordinatae BC, et lineae curvae
AO augmenta modo genita erunt B£, E^, et Cc ; et
in horum augmentorum nascentium ratione prima
sunt latera trianguli GET, ideoque fluxiones ipsarum
AB, BC et AC sunt ut trianguli illius CET latera
CE, ET et CT et per eadem latera exponi possunt,
vel quod perinde est per latera trianguli consimilis
VBC.

30. "Eodem recidit si sumantur fluxiones in
ultima ratione partium evanescentium. Agatur
recta Cc et producatur eadem ad K. Redeat ordinata
be in locum suum priorem BC, et cceuntibus punctis
C et r, recta CK coincidet cum tangente CH, et
triangulum evanescens CE^ in ultima sua forma
evadet simile triangulo CET, et ejus latera evanes-
centia CE, E^ et Cc erunt ultimo inter se ut sunt
trianguli alterius CET latera CE, ET et CT, et
propterea in hac ratione sunt fluxiones linearum
AB, BC et AC. Si puncta C et c parvo quovis
intervallo ab invicem distant recta CK parvo inter-
vallo a tangente CH distabit. Ut recta CK cum
tangente CH coincidat et rationes ultimae linearum
CE, E<: et Cr inveniantur, 'debent puncta C et c
coire et omnino coincidere. jLrrores quam minimi
in rebusmathematicis non sunt cpntpmnendif

31. " Simili argumento si circulus centre B radio
BC descriptus in longitudinem abscissae AB ad

20 LIMITS AND FLUXIONS

angulos rectos uniformi cum motu ducatur, fluxio
solidi geniti ABC erit ut circulus ille generans, et
fluxio superficial ejus erit ut perimeter circuli illius
et fluxio lineae curvae AC conjunctim. Nam quo
tempore solidum ABC generatur ducendo circulum
ilium in longitudinem abscissae AB, eodem super-
ficies ejus generatur ducendo perimetrum circuli illius
in longitudinem curvae AC. . . .

32. " Flu at quantitas x uniformiler et inveniendct
sit fluxio quantitatis xn. Quo tempore quantitas x
fluendo evadit x-\-o, quantitas xn evadet x-\-o\n, id
est per methodum serierum infinitarum, xn-\-noxH~l
-\-(nn — n)l 2 0<?;tr*~2 + etc. Et augmenta o et noxn~l
+ (nn — n) I 2 0ar*~2 + etc. sunt ad invicem ut I et nxn~*
+ (nn — n)/2 oxn~2 + etc. Evanescant jam augmenta
ilia, et eorum ratio ultima erit I ad nxn~^ : ideoque
fluxio quantitatis x est ad fluxionem quantitatis xn
ut i ad nxn~l.

33. "Similibusargumentis per methodum rationum
primarum et ultimarum colligi possunt fluxiones
linearum seu rectarum seu curvarum in casibus
quibuscunque, ut et fluxiones superficierum, angu-
lorum et aliarum quantitatum. In finitis autem
quantitatibus Analysin sic instituere, et finitarum
nascentium vel evanescentium rationes primas
vel ultimas investigare, consonum est geometriae
veterum : et volui ostendere quod in Methodo
Fluxionum non opus sit figuras infinite parvas
in geometriam introducere. Peragi tamen potest
Analysis in figuris quibuscunque seu finitis seu
infinite parvis quae figuris evanescentibus finguntur

NEWTON 21

similes, ut et in figuris quae pro infinite parvis
haberi solent, modo caute procedas. "

Translation by John Stewart :

' * Introduction

34. "I consider mathematical quantities in this
place not as consisting of very small parts ; but
as describ'd by a continued motion. Lines are
describ'd, and thereby generated not by the appo-
sition of parts, but by the continued motion of
points ; superficies's by the motion of lines ; solids
by the motion of superficies's ; angles by the rota-
tion of the sides ; portions of time by a continual
flux : and so in other quantities. These geneses
really take place in the nature of things, and are
daily seen in the motion of bodies. And after this
manner the ancients, by drawing moveable right
lines along immoveable right lines, taught the
genesis of rectangles.

35. " Therefore considering that quantities, which
increase in equal times, and by increasing are
generated, become greater or less according to the
greater or less velocity with which they increase
and are generated ; I sought a method of determin-

^ quantities from the velocities of the motions
or increments, with which they are generated ; and
calling these velocities of the motions or increments
Fluxions, and the generated quantities Fluents , I
fell by degrees upon the Method of Fluxions, which
I have made use of here in the Quadrature of
Curves, in the years 1665 and 1666.

22 LIMITS AND FLUXIONS

36. "Fluxions are very nearly as the augments
of the fluents generated in equal but very small
particles of time, and, to speak accurately, they are
in the first ratio of the nascent augments ; but they
may be expounded by any lines which are pro-
portional to them.

37. "Thus if the area's ABC, ABDG be
described by the ordinates BC, BD moving along
the base AB with an uniform motion, the fluxions of
these area's shall be to one another as the describ-
ing ordinates BC and BD, and may be expounded by
these ordinates, because that these ordinates are as
the nascent augments of the area's.

38. "Let the ordinate BC advance from it's
place into any new place be. Complete the par-
allelogram BCE£, and draw the right line VTH
touching the curve in C, and meeting the two lines
be and BA produc'd in T and V : and B^, E^ and
C<: will be the augments now generated of the
absciss AB, the ordinate BC and the curve line
AC<: ; and the sides of the triangle CET are in the

first ratio of these augments considered as nascent,
therefore the fluxions of AB, BC and AC are as
the sides CE, ET and CT of that triangle CET,
and may be expounded by these same sides, or,
which is the same thing, by the sides of the triangle
VBC, which is similar to the triangle CET.

39. "It comes to the same purpose to take the
fluxions in the ultimate ratio of the evanescent
parts. Draw the right line C<r, and produce it to
K. Let the ordinate be return into it's former

NEWTON 23

place BC, and when the points C and c coalesce,
the right line CK will coincide with the tangent
CH, and the evanescent triangle CErin it's ultimate
form will become similar to the triangle GET, and
it's evanescent sides CE, Er and Cc will be ulti-
mately among themselves as the sides CE, ET and
CT of the other triangle GET, are, and therefore
the fluxions of the lines AB, BC and AC are in the
same ratio. If the points C and c are distant from
one another by any small distance, the right line
CK will likewise be distant from the tangent CH
by a small distance. That the right line CK may
coincide with the tangent CH, and the ultimate
ratios of the lines CE, *Ec and Cc may be found, the
points C and c ought to coalesce and exactly co-
incide. The very smallest errors in mathematical
matters are not to be neglected.

40. "By the like way of reasoning, if a circle
describ'd with the center B and radius BC be drawn
at right angles along the absciss AB, with an uni-
form motion, the fluxion of the generated solid
ABC will be as that generating circle, and the
fluxion of it's superficies will be as the perimeter of
that circle and the fluxion of the curve line AC
jointly. For in whatever time the solid ABC is
generated by drawing that circle along the length
of the absciss, in the same time it's superficies is
generated by drawing the perimeter of that circle
along the length of the curve AC. ..."

41. "Let the quantity x flow uniformly, and let it
be proposed to find the fluxion of xn.

24 LIMITS AND FLUXIONS

"In the same time that the quantity ;tr, by
flowing, becomes x-\-o, the quantity xn will
become x+o\n, that is, by the method of infinite
series's, xn-\-noxn~ljf(n2 — n)J2 o oxn ~2 + etc. And
the augments o and noxn~l-{-(rtz — n)/2 ooxn~'1 + etc.
are to one another as I and nxn~l-)-(n2 — n)/2 oxn~z
-fete. Now let these augments vanish, and their
ultimate ratio will be I to nxn~l.

42. "By like ways of reasoning, the fluxions of
lines, whether right or curve in all cases, as likewise
the fluxions of superficies's angles and other quan-
tities, may be collected by the method of prime and
ultimate ratios. Now to institute an analysis after
this manner in finite quantities and investigate the
prime or ultimate ratios of these finite quantities
when in their nascent or evanescent state, is con-
sonant to the geometry of the ancients : and I was
willing to show that, in the Method of Fluxions,
there is no necessity of introducing figures infinitely
small into geometry. Yet the analysis may be
performed in any kind of figures, whether finite or
infinitely small, which are imagin'd similar to the
evanescent figures ; as likewise in these figures,
which, by the Method of Indivisibles, ^used to be
reckoned as infinitely small, provided you proceed
with due caution."

43. In the Quadrature of Curves proper, under
"Proposition I" the proof of the rule for finding
the fluxion of expressions like x* — xy^ + a^z — £3 = o
contains the following passages which indicate the
use made of the symbol "o" and of the term

NEWTON 25

41 moment," and the mode of passing to the limit.
We quote : —

' * Demonstratio

44. "Nam sit o quantitas admodum parva et
sunto oz, oy, ox, quantitatum z, y, x, momenta id
est incrementa momentanea synchrona. Et si
quantitates fluentes jam sunt z, y et x, hae post
momentum temporis incrementis suis oz, oy, ox
auctae, evadent z + oz, y + oy, x+ox, quae in
aequatione prima pro zt y et x scriptae dant
aequationem . . .

$xx2 + $xxox+ x*oo — xyy — 2xyy

— 2xoyy — xoyy — xooyy + aaz = o.

Minuatur quantitas o in infmitum, et neglectis
terminis evanescentibus restabit $xx2 — xyy— 2xyy
z = o. Q.E.D."

Translation by John Stewart :

" Demonstration

45. " For let o be a very small quantity, and let
ozt oy, ox be the moments, that is the momentaneous
synchronal increments of the quantities z, y, x.
And if the flowing quantities are just now z, y, x,
then after a moment of time, being increased by
their increments oz, oy, ox\ these quantities shall
become z + oz, y + oy, x-\-ox\ which being wrote
in the first equation for z, y and x, give this
equation . . .

3^-^ -f -$xxox + i?oo — xyy — 2xyy

— 2xoyy — xoyy — xooyy -\- aaz = o.

26 LIMITS AND FLUXIONS

Let the quantity o be diminished infinitely, and
neglecting the terms which vanish, there will remain
^xx* — xyy — 2xyy + aaz = o. Q.E. D. "

(^te&tCt/ Jk//f/\jt^^

IV. AN ACCOUNT OF THE "COMMERCIUM
EPISTOLICUM "

46. It is now generally accepted that the account l
of the Commercium Epistolicum, published in the
Philosophical Transactions, London, 1717, was
written by Newton. The reasons for attributing it
to him are stated by De Morgan2 and by Brewster.3
In abstract the account is as follows : —

47. (Pp. 177-178.) In a letter of October 24,
1676, to Oldenburgh, Newtoji explained that in
deducing areas he considered the area as growing
"by continual Flux"; "from the Moments of Time
he gave the Name of Moments to the momentaneous
Increases, or infinitely small Parts of the Abscissa
and Area generated in Moments of Time. The
Moment of a Line he called a Point, in the Sense
of Cavalerius, tho' it be not a geometrical Point,
but a Line infinitely short, and the Moment of an
Area or Superficies he called a Line, in the sense
of Cavalerius, tho' it be not a geometrical Line,

1 Philosophical Transactions, vol. xxix, for the years 1714, 1715,
1716. London, 1717. " An Account of the Book entituled Commer-
cium Rpistolicum Collinii et aliorum, De Analyst promota . . .,"
pp. 173-224. This account was translated into Latin and inserted in
the edition of the Commercium Rpistol^c^lm of 1725.

2 See De Morgan's articles in the Philosophical Magazine, S. 4,
vol. iii, June, 1852, pp. 440-444 ; vol. iv, November 1852, p. 323.

3 Sir David Brewster, Memoirs of the Life, Writings, and Discoveries
of Sir Isaac Newton, 2nd ed., vol. ii, Edinburgh, 1860, pp. 35, 36.

NE WTON 27

but a Superficies infinitely narrow. And when he
consider'd the Ordinate as the Moment of the Area,
he understood by it the Rectangles under the geo-
metrical Ordinate and a Moment of the Abscissa,
tho' that Moment be not always expressed. " Again,
p. 179: "And this is the Foundation of the
Method of Fluxions and Moments, which Mr.
Newton in his Letter dated Octob. 24, 1676,
comprehended in this Sentence. Data cequatione
quotcunque fluent es quantitates involvente, invenire
Fluxiones ; et vice versa. In this Compendium
Mr. Newton represents the uniform Fluxion of
Time, or of any Exponent of Time by an Unit ;
the _ Moment of Time or its Exponent by the Lettej;
'o\ the Fluxions of other Quantities by any other
Symbols ; the Moments of those Quantities by the
Rectangles under those Symbols and the Letter o ;
and the Area of the Curve by the Ordinate inclosed
in a Square, the Area being put for a Fluent and
the Ordinate for its Fluxion. When he is demon-
strating a Proposition he uses the Letter o for a
finite Moment of Time, or of its Exponent, or of
any Quantity flowing uniformly, and performs the
whole Calculation by the Geometry of the Ancients
in finite Figures or Schemes without any Approxi-
mation : and so soon as the Calculation is at an
End, and the Equation is reduced, he supposes that
the moment o decreases in infinitum and vanishes.
But when he is not demonstrating but only investi-
gating a Proposition, for making Dispatch he
supposes the Moment o to be infinitely little, and

28 LIMITS AND FLUXIONS

forbears to write it down, and uses all manner of
Approximations which he conceives will produce

/^no Error in the Conclusion." In Newton's Princifiia
" he frequently considers Lines as Fluents described

' by Points, whose Velocities increase or decrease,
the Velocities are the first Fluxions, and their
Increase the second." The Compendium of his
Analysis was written "in or before the year 1669"
(p. 1 80). "And the same Way of working he used
in his Book of Quadratures, and still uses to this
day "(p. 182). On p. 204 we read : "Mr. Newton
used the letter o in his Analysis written in or before
the Years 1669, and in his Book of Quadratures,
and in his Principia Philosophic?, and still uses it
in the very same Sense as at first. . . . These
Symbols o and x are put for things of a different kind.
The one is a Moment, the other a Fluxion or
Velocity as has been explained above. . . . Prickt
Letters never signify Moments, unless when they
are multiplied by the Moment o either exprest or
understood to make them infinitely little, and then
the Rectangles are put for the Moments " (p. 204).
Further on we read: "It [the method of fluxions]
is more elegant [than the Differential Method of
Leibniz], because in his Calculus there is but one
infinitely little Quantity represented by a symbol,
the symbol o. We have no Ideas of infinitely little
Quantities, and therefore Mr. Newton introduced
Fluxions into his Method, that it might proceed by
finite Quantities as much as possible. It is more
Natural and Geometrical, because founded upon the

NE WTON 29

primes quantitatum nascentium rationes, which have
a Being in Geometry, whilst Indivisibles, upon
which the Differential Method is founded, have no
Being either in Geometry or in Nature. There are
"rationes primce quantitatum nascentium, but not
quantitates primce nascentes. Nature generates
Quantities by continual Flux or Increase ; and the
ancient Geometers admitted such a Generation of
Areas and Solids " (p. 205).

From Newton's Correspondence and Manuscripts
not in print in 1734

48. Manuscripts of Newton, some of them still
unpublished, show that he first thought of fluents
and fluxions in 1665 and 1666, when he was in
his twenty-third and twenty-fourth years.1 The
notation by dots occurs as early as 1665. As
pointed out by De Morgan,2 these early papers are
infinitesimal in character. They were first published
in i838.3 A manuscript, dated Nov. 13, 1665,
gives rules for finding the velocities /, q, r, etc.,
of two or more lines x, y, z, etc. , described by bodies
A, B, C, etc., the lines being related to each other

1 See a list of Newton's manuscripts and publications on fluxional
calculus prepared by Philip E. B. Jourdain, in his edition of Augustus
De Morgan's Essays on the Life and Work of Newton, The Open
Court Publishing Co., 1914, pp. 107-112.

2 Augustus De Morgan, "On the Early History of Infinitesimals
in England," J^he London, Edinburgh, and Dublin Philosophical
Magazine, 4th S., vol. iv, 1852, pp. 321-330. This article is an
important historical contribution, of which extensive use is made in the
present history.

3 See S. P. Rigaud, Historical Essay on the first Publication of Sir
Isaac Newton s Principia, Oxford, 1838, Appendix, pp. 20-24.

30 LIMITS AND FLUXIONS

by an equation, such as x* — zcfiy + zzx — yyx + zyy —
£3 = o. "If the body A, with the velocity /,
describe the infinitely little line o in one moment,
in the same moment B, with the velocity q, will
describe the line oq / /," and the body C, with the
velocity r, will describe the line or / p. So that,
if the described lines be x, y and z "in one
moment," they will be x + o, y + oq / /, 2 + or / p
"in the next." He finds that the relation of the
velocities p, q, r, in the above example, is $pxx +
pzz —pyy — 2aaq — 2yxq + 2zyq + 2zxr +yyr — ^zzr = o.
In proving his rules for differentiation, Newton
divides by o, and in the resulting expression observes
that "those terms in which o is, are infinitely less
than those in which it is not. Therefore, blotting
them out, there rests " the relation sought. The
notation by dots, " pricked letters," occurs on a leaf,
dated May 20, 1665, which has never been printed.1

49. It is evident that Newton permitted twenty-
eight years to pass between the time of his first
researches on fluxions and 1693, the date when the
earliest printed account of his notation of fluxions
appeared from his pen in the Latin edition of Wallis's
Algebra. Moments and fluxions are mentioned in
his Principia, as has been shown by our extracts.

50. Of importance in the interpretation of the
meanings of "moment" in the second edition of

1 S. P. Rigaud, op. cit.t Appendix, p. 23. Consult also the remarks
on this passage made by G. Enestrom in Bibliotheca viathematica,
3. F., Bd. n, Leipzig, 1910-1911, p. 276, and Bd. 12, 1911-1912,
p. 268, and by A. Witting in Bd. 12, pp. 56-60. See also A catalogue
of the Portsmouth collection of books and papers, written by or belonging
to Isaac Newton, Cambridge, 1889.

NEWTON 31

the Principia (1713) is a letter of May 15, 1714,
from Newton to Keill,1 from which we quote the
following : —

51. ". . . altho I use prickt Letters in the first
Proposition of the book of Quadratures, yet I do
not there make them necessary to the method.
For in the Introduction to that book I describe
the method at large & illustrate it wth various
examples without making any use of such letters.
And it cannot be said that when I wrote that
Preface I did not understand the method of fluxions
because I did not there make use of prickt letters
in solving of Problems.2 The book of Quadratures
is ancient, many things being cited out of it by me
in my Letter of 24 Octob. 1676. . . .

52. "ffiuxions & moments are quantities of
a different kind, ffluxions are finite motions,
moments are infinitely little parts. I put letters
with pricks for fluxions, & multiply fluxions by the
letter o to make them become infinitely little and
the rectangles I put for moments. And wherever
prickt letters represent moments & are without the
letters this letter is always understood. Wherever
•*» y* y-> y> etc- > are Put f°r moments they are put for
jtt?, yo^ yoo, yo*. In demonstrating Propositions I
always write down the letter o & proceed by the
Geometry of Euclide and Apollonius without any

1 ]. Edleston, Correspondence of Sir Isaac Aewton and Professor
Coles, London, 1850, pp. 176, 177.

2 John Bernoulli, in the Acta Eruditorum for February and March,
1713} had criticised a passage in the Principia, and claimed that Newton
did not understand the second fluxions when writing that passage.

32 LIMITS AND FLUXIONS

approximation. In resolving Questions or investi-
gating truths I use all sorts of approximations
wch 1 think will create no error in the conclusion
and neglect to write down the letter 0, and this do
for making dispatch. But where x, j>, J/, y are put
for fluxions without the letter o understood to make
them infinitely little quantities they never signify
differences. The great Mathematician l therefore
acts unskilfully in comparing prickt letters with
the marks dx and dy, those being quantities of a
different kind. "

Remarks

53. The extracts from Newton's writings demon-
strate the following : —

(i) At first Newton used infinitesimals (infinitely
small quantities), as did Leibniz and other mathe-
maticians of that age. As early as 1665, when
Newton was a young man of twenty-three, he used
them and speaks of "blotting them out."2 He
uses infinitesimals in the Principia of I68/3 and in
his account of the quadrature of curves in Wallis's
Algebra of 1693, where Newton speaks of himself
in the third person.4 It is worthy of emphasis, in
contrast to Leibniz, that Newton uses only infinitesi-
mals of the first order. Moreover, as De Morgan
remarked long ago,5 "the early distinction between
the systems of the two is this, that Newton,
holding to the conception of the velocity or fluxion,

1 John Bernoulli. See Edleston, op. cit.> p. 171. 2 See our § 48.
3 See our §§ 10, 13, 16, 18. 4 See our §§ 21, 26.

5 De Morgan, Philosophical Magazine, 4 S., vol. iv, 1852, p. 324.

NEWTON 33

used the infinitely small increment as a means of
determining it ; while, with Leibnitz, the relation
of the infinitely small increments is itself the object
of determination. "

(2) As early as 1665, Newton speaks of describing
an "infinitely little line" in "one moment," and
then uses the expression "in the next" moment.1
Here "moment" cannot mean a point of time,
destitute of duration ; it means an infinitely small
duration, an infinitesimal of time. Doubtless this
use of "moment" with reference to time suggested
the more extended and general use of the term
"momentum" or "momenta" as found in the
Principia'1 and later publications.

(3) The use of dots, " prickt letters," to indicate
velocities or fluxions goes back to 1665, 3 but they
are not used by Newton in print until 1693 in
Wallis's Algebra ; they are used extensively in
Newton's Quadrature of Curves of 1704.*

(4) Newton first used the word * * fluxion " In
print in 1687 in the Frincipia*

(5) The first refinement of the doctrine of fluxions
(is found in Newton's Principia, where he speaks of
^ grime and ultimate ratios"6 and of "limits*." 7

(6) The high-water mark of Newton's efforts to
place the doctrine of fluxions upon a thoroughly
logical basis is found in his Quadrature of Curves,
1704. It indicates the almost complete exclusion

1 See our § 48. 2 See our §§ 16, 18, 21, 24.

3 See our § 48. * See our §§ 22, 25, 44, 45.

5 See our §§ 16, 18. 6 See our §§ 10, 13.

7 See our §§ 4, 6, 8, 10, 13.

34 LIMITS AND FLUXIONS

of quantities infinitely little. <c ^consider mathe-
matical quantities in this place not as consisting of
very small parts," says Newton.1 Also "the very
smallest errors in mathematical matters are not to
be neglected, " 2 and * ' in the method of fluxions
there is no necessity of introducing figures infinitely
small into geometry."3 In view of these statements
the syinbpj o used-iu. the Quadrature of Curves^ a
' ' quanjjlas-ad^mo^m parya^" 4 ^_mu_st_be_interpreted
as a small finite quantity. In this connection De
Morgan's remarks are of interest:5 "In 1704,
Newton in the Quadratura Curvarum renounced
and abjured the infinitely small quantity ; but he
did it in a manner which would lead any one to
suppose that he had never held it. ... And yet,
there is something like a recognition of some one
having used infinitely small quantities in Fluxions,
contained in the following words : volui ostendere
quod in Methodo Fluxionum non opus sit figuras
infinite parvas in Geometriam introducere : nothing
is wanted except an avowal that the some one was
Newton himself. The want of this avowal was
afterwards a rock of offence. Berkeley, in the
Analyst, could not or would not see that Newton
of 1687 and Newton of 1704 were of two different
modes of thought. "

We do not interpret Newton's expressions of
1704 as declarations that a logical exposition of

1 See our §§ 27, 34. 2 See our §§ 30, 39.

3 See our §§ 33, 41. 4 See our §§ 44, 45.

5 De Morgan, Philosophical Magazine, 4 S. , vol. iv. 1852,
p. 328.

NEWTON 35

fluxions cannot be given on the basis of infini-
tesimals or that infinitely small quantities are
impossible; for he says,1 "the analysis may be
performed in any kind of figures whether finite or
infinitely small, which are imagined similar to the
evanescent figures."

In fact, not even in 1704 did Newton succeed in
completely banishing from his doctrine of fluxions
the infinitely little. If what he used in 1704 is
not the infinitely Iittle1 it is so <flnsely related thereto , \
that it cannot be called either a finite magnitude or
an absolute zero.

In 1704, fluxions are "in the first ratio of the
nascent augments," or "in the ultimate ratio of the
evanescent parts. " 2 Unless the fully developed
theory of limits is read into these phrases, they
will involve either infinitely little parts or other
quantities no less mysterious. At any rate, the
history of fluxions shows that these expressions
did not meet the demands for clearness and freedom
from mysticism. Newton himself knew full well
the logical difficulty involved in the words "prime
and ultimate ratios"; for in 1687 he said,3 "it is
objected, that there is no ultimate proportion of
evanescent quantities ; because the proportion,
before the quantities have vanished, is not ultimate;
and, when they have vanished, is none." How does
Newton meet this, his own unanswerable argument ?
He does so simply by stating the difficulty in another

1 See our §§ 33, 42. 2 See our §§ 29, 30, 33, 36, 38, 39, 42.
3 See our §§ 11, 14.

36 LIMITS AND FLUXIONS

form: "But, by the same argument, it might as
well be maintained, that there is no ultimate velocity
of a body arriving at a certain place, when its motion
is ended : because the velocity, before the body
arrives at the place, is not its ultimate velocity ;
when it has arrived, is none. But the answer is
easy : for by the ultimate velocity is meant that
... at the very instant when it arrives." If
" instant," as used here, is not an infinitesimal, the
passage would seem to be difficult or impossible of
interpretation.

(7) A return to the open use of the infinitely
small quantities is seen in writings of Newton after
the year 1704. It might be argued that such a

) return was necessary in the second edition of the
Principia, 1713, unless the work were largely re-
written. Newton's Analysis per cequationes numero
terminorum infinitas was first printed in 1711, and
might have been rewritten so as to exclude infini-
tesimals as fully as was done in the Quadrature of
Curves of 1704. But the infinitely little is per-
mitted to remain.1 There is no disavowal of such
quantities either in the Commercium Epistolicum,
with the editors of which Newton was in touch,
or in Newton's own account of this publication,
contributed to the Philosophical Transactions*

(8) The theory of limits is involved in the first
lemma of the Principia* and in the explanation of
prime and ultimate ratios as given in that work.

1 See our § 66. 2 See our § 47.

3 See our §§ 4, 6, 8, 9, 10, 12, 13, 15.

CHAPTER II .

PRINTED BOOKS AND ARTICLES ON FLUXIONS
BEFORE 1734

54. THE earliest printed publication in Great
Britain on the new calculus was from the pen of
John Craig, a Scotsman by birth, who settled in
Cambridge and became a friend of Newton. Later
he was rector of Gillingham in Dorsetshire. He
was "an inoffensive, virtuous man," fond of mathe-
matics. In 1685 ne published at London a book
entitled, Methodus figurarum . . . quadraturas
determinandi. At that time nothing could be
known about fluxions except through private com-
munication. In 1684 Leibniz published his first
ideas of Differential Calculus in the Leipzig Acts.
Craig used in 1685 the calculus of Leibniz and also
the notation of Leibniz. Continental writers call
Craig the introducer of the_ theory of Leibniz into
England. On p. 28 of his book, Craig derives
dp — 4&nr*yzdy from fl=i6nr4yB, and arrives at a
differential equation (sequationem differentialem).
The meanings of dp, dy, dx, etc., are not explained
but taken for granted, reference being made to
Leibniz. In 1693 Craig published another book in
which the notation of Leibniz is used. He con-

38 LIMITS AND FLUXIONS

tributed also several papers to the Philosophical
Transactions (London), but never, before 1718, did
he use fluxional symbols. In preparing the book
of 1685 he had received from Newton the binomial
theorem which he used before it had appeared in
print, but he had no communication about fluxions.
" We have here the singular indifference," says De
Morgan, "which Newton at that time, and long
afterwards, showed toward his own calculus."1
Craig wrote a tract in 1693, and articles for the
Philosophical Transactions in 1701, 1703, 1704, 1708,
using the differential calculus all this time. In the
issue No. 284, 1703, he employs the Leibnizian
sign of integration /. Craig submitted to Newton
one of his early manuscripts (probably the one
printed in 1693). With regard to this event De
Morgan wrote to Hamilton, the inventor of 'quater-
nions : (( Few of us know that Leibniz was perfectly
well known in England before the dispute, and that
Newton's first provocative to an imperfect publica-
tion was ds and infinitely small quantities paraded
under his own eyes by an English writer (Craig),
who lent him his MSS. to read."2 Craig's publica-
tion of 1718 followed the great controversy on the
invention of the calculus ; now he uses fluxions
exclusively and says not a word on the differential
calculus. The book does not discuss fundamentals,
and no explanation of x is given. As conjectured

1 De Morgan, Philosophical Magazine, 4 S. , vol. iv, 1852, p.
326.

2 Life of Sir William Rowan Hamilton, by Robert P. Graves,
vol. iii, 1889, p. 415.

PRINTED BOOKS, ETC., BEFORE 1734 39

by De Morgan, it may have been Craig's manu-
script that suggested to Newton the need of making
his own fluxions accessible to the public. At any
rate, in 1693 there appeared the account of fluxions
in Wallis's Algebra. [See Addenda, p. 289.]

55. Abraham De Moivre, a French mathemati-
cian who in 1688, after the revocation of the Edict
of Nantes, came to London, contributed in 1695 to
No. 216 of the Philosophical Transactions (London)
an article in which he uses x, y, x, y, and lets both
"fluxion" and "moment" stand for things infin-
itely small. In the same number of the Transac-
tions, the astronomer Edmund Halley has an article
on logarithms in which he uses infinitely small
ratiunculce and differentiolce, but neither the nota-
tion of Leibniz nor that of Newton. In 1697, David
Gregory used in No. 231 of the Transactions x and
speaks of " fluxio fluxionis " without, however, ex-
plaining his terms.

56. Fatio de Duillier, a Swiss by birth, who had
settled in London and become member of the
Royal Society, wrote in 1699 a treatise, Line<z
brevissimi descensus investigatio geometrica, uses
fluxions as infinitely small quantities. This publi-
cation is noted as containing a statement which
started the Newton-Leibniz controversy on the
invention of the Calculus.

57. It is remarkable that Roger Cotes, in 1701,
when an undergraduate at Trinity College, Cam-
bridge [Newton's own College], wrote a letter on
mathematical subjects, in which x is used as

40 LIMITS AND FLUXIONS

4 'infinitely little."1 In 1702-3 Humphry Ditton,
in vol. xxiii of the Transactions, used the fluxional
notation, without explanation.

58. Other writings that do not define their terms
are the Fluxionum methodus inversa, 1704, by the
London physician, George Cheyne, and De Moivre's
Animadversions in D. Georgii Cheynai Tractatum,
London, 1704. However, Cheyne lets £= I, from
which we infer that, with him, x was finite. [See
Addenda, p. 289.]

59. The next writer on fluxions was John Harris,
a v&uminous author of books on various subjects.
He was at one time Secretary of the Royal Society.
In 1702 he published at London A New Short
Treatise of Algebra, which devotes the last 22
pages, out of a total of 136 pages, to fluxions. It
is the first book in the English language in which
this subject is treated. The doctrine of fluxions is
the " Arithmetick of the Infinitely small Increments
or Decrements of Indeterminate or Variable Quan-
tities, or as some call them the Moments or Infin-
itely small Differences of such Variable Quantities.
These Infinitely small Increments or Decrements,
our incomparable Mr. Isaac Newton calls very pro-
perly by this name of Fluxions " (p. 115). A few
lines further on it says that Newton "calls the
celerity or Velocity of the Augmentation o£ Diminu-
tion of these Flowing Quantities, by the name of
Fluxions." A second edition of this book appeared
in 1705. As authors on fluxions, Harris in 1705

1 J. Edleston, Correspondence of Sir Isaac Newton and Professor
Cotes t London, 1850, p. 196.

PRINTED BOOKS, ETC., BEFORE 1734 41

mentions Newton, Wallis, Nieuwentiit, Carre, Leib-
niz, 1'Hospital, de Moivre, and Hayes.

60. John Harris also published a Lexicon Tech-
nicum, of which the second volume, London, 1710,
contains an article, "Fluxions."

"This general Method of finding the Fluxions of
all Powers and Roots, I had from the Hon. Fr.
Robartes, Esq. If a Quantity gradually increases
or decreases, its immediate Increment or Decre-
ment is called its Fluxion, Or the Fluxion of
a Quantity is its Increase or Decrease indefinitely

(small. . . . Since xx ... is infinitely smaller than
2xx, whereby it can make no sensible Change in
that Quantity, it may be laid aside as of no Value.
. . . Authors' Names who have written of Fluxions:
D. Bernoulli Tractatus de Principiis Calculi Exponen-
tialis\ Nieuwentiifs Analysis Infinitorum, Amster.,
1695 ; Dr. Cheyne's Fluxions, with Moivre 's Anim-
adversions on them, and the Doctor's reply ; Hays' s
Fluxions, Lond. , 1704; Analyse des Infiniment
Petits. Part' Hospital, Fr., Paris, 1696; Le Calcule
Integrate, par M. Carre, Paris, 1700; Mr. Abraham
de Moivre's Use of Fluxions, in the Solution of
Geometrick Problems. See P kilos. Trans., N. 216;
Mr. Humphry Dittoris Institution of Fluxions. "

6 1. In the above list of writers are Charles
Hayes and Humphry Ditton, authors of English
texts now demanding our attention. Hayes starts
his elucidation of fundamentals (p. i) as follows -,1

1 A Treatise of Fluxions: or, An Introduction to Mathematical
Philosophy, Charles Hayes, London, 1704.

42 LIMITS AND FLUXIONS

' ' Magnitude is divisible in infinitum, and the
Parts after this infinite Division, being infinitely
little, are what Analysts call Moments or Differ-
ences ; And if we consider Magnitude as Indeter-
minate and perpetually Increasing or Decreasing,
then the infinitely little Increment or Decrement is
call'd the Fluxion of that Magnitude or Quantity :
And whether they be called Moments, Differences
or Fluxions, they are still suppos'd to have the
same Proportion to their Whole's, as a Finite
Number has to an Infinite ;
or as a finite Space has to
an infinite Space. Now those
infinitely little Parts being
extended, are again infinitely
Divisible ; and these infinitely
little Parts of an infinitelylittle
FlG Part of a given Quantity, are

by Geometers call'd Infinite -

siuice Infinitesimarum or Fluxions of Fluxions.
Again, one of those infinitely little Parts may be
conceiv'd to be Divided into an infinite Number of
Parts which are call'd Third Fluxions, etc. "

He endeavours to justify this doctrine by illus-
trations. The angle of contact FAC formed by the
line AE and the ordinary parabola AC, is less than
any rectilineal angle ; the angle FAD, formed by
AE with the cubical parabola AD, is infinitely
less than the angle FAC, and so on. Hayes
defines the doctrine of Fluxions as the ' ' Arith-
metick of infinitely small Increments or Decrements

PRINTED BOOKS, ETC., BEFORE 1734 43

of Indeterminate or variable Quantities." He
cautions the reader: "But we must take great
heed, not to consider the Fluxions, or Increments,
or Decrements as finite Quantities " (p. 4). He
rejects xzy and xzy ' ' as being incomparably less "
than xzy.

The same year in which Hayes wrote this first
English book on fluxions which could make any
claim to attention, saw the appearance of Newton's
Quadratura Curvarum. The contrast in the defini- '
tion of "fluxion" was sharp. Hayes called it "an
infinitely small increment " ; Newton called it a
"velocity," a finite quantity.

62. William Jones, in his Synopsis Palmariorum
Matties eos, London, 1706, devotes a few pages to
fluxions and fluents, using the Newtonian notation.
On p. 225 he gives, in substance, Newton's lemma,
in these words: "Quantities, as also their Ratio's,
that continually tend to an Equality, and therefore
that approach nearer the one to the other, than
any Difference that can possibly be assign'd, do
at last become equal." Then he says : "Hence all
Curved Lines may be considered as composed of
an Infinite Number of Infinitely little right Lines."
He uses "infinitely small" quantities, but defines
a fluxion as "the Celerity of the Motion," fluxions
being "in the first Ratio of their Nascent Aug-
ments." Jones represents here the Newton of the
Principia, and of the Quadrature of Curves as given
in 1793.

63. The earliest book exhibiting a careful study

44 LIMITS AND FLUXIONS

of Newton's tract of 1704 was Humphry Ditton's
Institution of Fluxions, 1706. * Ditton was pro-
minent as a divine as well as a mathematician.
Like so many other English writers on fluxions
during the eighteenth century, he had not been at
either of the great universities. He states in his
preface that he has also consulted and drawn from
the writings of John Bernoulli and some other
Continental writers.

64. The reader of Ditton's book is impressed by
the fact that he labours strenuously to make every-
thing plain. He takes the reader fully into his
confidence. This is evident in the extracts which
follow (pp. 12-2 1): —

''Suppose any flowing Quantities, ... as also
their Increments . . . which Increment imagine to
be generated in equal very small Particles of Time.
I conceive we may say without Scruple, that the
Fluxions are the velocities of those Increments, con-
sider'd not as actually generated, but quatenus
Nascentia, as arising and beginning to be generated.
As there is a vast difference between the Increments
considered as Finite, or really and actually generated ;
and the same considered only as Nascentia or in the
first Moment of their Generation : So there is as
great a difference also between the Velocities of the
Increments, consider'd in this two fold respect. . . .

1 An Institution of Fluxions : Containing the First Principles , The
Operations, with some of the Uses and Applications of that Admirable
Method ; According to the Scheme prefix d to his Tract of Quadratures,
by (its First Inventor} the Incomparable Sir Isaac Newton. By
Humphry Ditton, London, 1706.

PRINTED BOOKS, ETC., BEFORE 1734 45

The Reason of that [difference], is this. Because
there is (speaking strictly and accurately) an Infinity
of Velocities to be consider'd, in the Generation and
Production of a Real Increment ; ... So that if we
conceiv'd the Fluxion, to be the Velocity of the
Increment, as actually Generated ; we must conceive
it to be an Infinite Variety or Series of Velocities.
Whereas the Velocity, with which any sort of In-
crement arises, or begins to be generated ; is a thing
that one may form a very clear and distinct Idea of,
and leaves the Mind in no Ambiguity or Confusion
at all. . . . However, if we take those Particles
of time exceeding small indeed, and Neglect the
Acceleration of the Velocity as inconsiderable, we
may say the Fluxions are proportional to those In-
crements ; remembering at the same time, that they
are but nearly, and not accurately so. ... If in the
Differential Calculus, some Terms are rejected and
thrown out of an Equation, because they are nothing
Comparatively, or with respect to other Terms in the
same Equation ; that is, because they are infinitely
small in proportion to those other Terms, and so may
be neglected upon that Score: On the other hand, in
the Method of Fluxions, those same Terms go out
of the Equation, because they are multiplied into a
Quantity, which . . . does at last really vanish. . . .
N. B. Speaking here of Infinitely small Quantities,
or Infinitesimals as some Authors (and particularly
Mr. Neiwentiit) chuse to term them, I cannot but
take notice of a notion, which that Excellent and In-
genious Person advances in. his Analysis Infinitorum.

46 LIMITS AND FLUXIONS

It is this ; That a Quantity Infinitely Great, a Finite or
any given Quantity, an Infinitesimal, and Nihilum
Geometricum, are in Geometrical Proportion. I
confess I cannot discover the truth of this. . . .
Let m denote an infinite Quantity, d any finite one ;
then is d / m the Infinitesimal of d, according to
Mr. Neiwentiit. Now his Assertion is, that m : d : :
d / m : o ; therefore since from the nature of Geo-
metrical Proportion, 'tis also m : d : : d j m : dd j mm ;
it follows that dd / mm is = o . . . then d / m = o.
Now Mr. Neiwentiit will hardly allow his Infinitesimal
to be nothing ; and yet ... I think it must follow,
that d=o." Proceeding geometrically, Ditton ex-
plains the fluxions of lines, areas, solids, and surfaces.
Next he takes up algebraical expressions. To find
the fluxion of xn, he lets x flow uniformly and re-
presents the augment of x in a given particle of time
by the symbol o. While x becomes x-\-o, xn becomes
(x+o)n. Expanding the binomial, he finds that the
two augments are as I to nxn~l-]-(n2' — n)oxn~2 / 2-f-
etc. " And the Ratio of them (making o to vanish)
will be that of I to nxn~l." According to his nota-
tion x is a fluxion of x, and x is a fluxion of x.
Taking o as a very small quantity, he lets the ex-
pressions ox, oy represent the moments, or increments
of the flowing quantity z, y generated in a very small
part of time. "If therefore now, at the present
Moment, the flowing Quantities are z, y, x\ the next
Moment (when augmented by these Increments)
they will become x+os, y + oy, x+ox" He ex-
presses the general mode of procedure for finding

PRINTED BOOKS, ETC., BEFORE 1734 47

the fluxion, which coincides with the modern mode
of finding a derivative. Ditton considers the in-
crements as finite (p. 53). "These Momenta are in
proportion to one another as the Fluxions of the
flowing Quantities respectively, for oz, oy, ox, are
as z, y, x\ and Mr. Newton had before expresly
told us ; that the Increments generated in a very small
Particle of time were very nearly, as the Fluxions."
Evidently Ditton does not here overlook that
oz, oy, ox represent the increments only "very
nearly." He observes (p. 98) that we may "go
on with ease to the second, third, and any other
Fluxions ; neither are there any new Difficulties to
be met with."

A second edition of Ditton's book was brought
out in 1726 by John Clarke.

65. Ditton's first edition appeared at a time when
the Newton-Leibniz controversy was under way.
Leibniz had appealed to the Royal Society for
justice. That Society appointed a committee which
published a report containing letters and other
material bearing on the case, in a book called the
Commercium Epistolicum,^- which figures prominently
in the lamentable controversy. From this book the
early use of infinitely small quantities on the part
of Newton is conspicuously evident. The book
makes it clear also that some of Newton's warmest
supporters were guilty of gross inaccuracy in the
use of the word ' ' fluxion. "

1 Commercium Epistolicum D. fohannis Collins, et aliortim de analyst
promota : jussu societatis regies in lucent editum, Londini, MDCCXII.

48 LIMITS AND FLUXIONS

66. Newton's Analysis per cequationes numero
terminorum infinitas, which was sent on July 31,
1669, through Barrow to Collins, and which was
first published at London in 1711, was reprinted in
the Commercium Epistolicum. In this Analysis in-
finitely small quantities are used repeatedly, but the
word "fluxion" and the fluxional notation do not
occur. In a letter to H. Sloane, who was then
Secretary of the Royal Society of London, written
in answer to a letter of Leibniz dated March 4, 17 1 1,
John Keill, professor of astronomy at Oxford, re-
counts the achievements of Isaac Barrow and James
Gregory, and says: "If in place of the letter o,
which represents an infinitely small quantity in
James Gregory's Geometric pars universalis (1667),
or in place of the letters a or e which Barrow em-
ploys for the same thing, we take the x or y of
Newton or the dx or dy of Leibniz, we arrive at the
formulas of fluxions or of the differential calculus."1
Thus Keill, the would-be great champion of Newton,
instead of warning the reader against confusing
differentials and fluxions, himself comes dangerously
close to conveying the erroneous idea that x and y
are infinitely small, the same as dx and dy. He
comes so near to this as to be guilty of lack of
caution, if not of inaccuracy.

More serious is a statement further on. The en-

1 "Nam si pro Litera o, quae in Jacobi Gregorii Parte Matheseos Uni-
versali quantitatem infinite parvam reprsesentat ; aut pro Literis a vel e
quas ad eandem designandam adhibet Barrovius ; ponamus x vel y
Newtoni, vel dx seu dy Leibnitii, in Formulas Fluxionum vel Calculi
Differ en tialis incidemus " (p. 112).

PRINTED BOOKS, ETC., BEFORE 1734 49

listment of the services of a clever lawyer would be
needed to acquit the editors of the Commercium
Epistolicum of gross error when, in the final summary
of their case against Leibniz, they declare (p. 121),
"that the Differential Method is one and the same
with the Method of Fluxions, excepting the name and
the notation ; Mr. Leibniz calling those Quantities
Differences, which Mr. Newton calls Moments or
Fluxions ; and marking them with the letter d, a
mark not used by Newton."

67. Joseph Raphson, in his History of Fluxions
(which appeared as a posthumous work at London,
in 1715, printed in English, and in the same year
also in Latin, the Latin edition containing new corre-
spondence bearing on the Newton-Leibniz contro-
versy), says on p. 5 that Newton ' * makes use of
Points, and denotes those first Differences (which by
a Name congruous to their Generation, being con-
sider'd as the first Increments or Decrements of a
continued Motion, he calls Fluxions] thus, viz. x, y,
s." This misrepresentation of Newton is the more
astonishing when we recollect that Raphson was
very partial to Newton, and also meant his History
"to open a plain and easy way for Beginners to
understand these Matters." Newton never looked
upon a fluxion as anything different from velocity ;
with him it was always a finite quantity. To make /
matters worse, "Raphson continues: "To these
Quantities he adds others of another Gender, and
which in relation to Finite ones may be conceiv'd as
infinitely great, and denotes them thus 'x, 'y, 'z,

50 LIMITS AND FLUXIONS

whereof the first or finite Quantities themselves, viz.
x, y, z, may be conceiv'd as Fluxions." And again,
"a Point . . . may be consider'd as the Fluxion of
a Line, a Line as the Fluxion of a Plane, and a
Plane as the Fluxion of a Solid, and a finite Solid
as the Fluxion of a (partially) infinite one, and that
again as the Fluxion of one of an higher Gender of
Infinity, and so on ad inf. which we shall further
illustrate in some Dissertations at the end of this
Treatise. "

68. Brook Taylor brought out at London in 1715
his Methodus incrementorum directa et inversa, in
which he looks upon fluxions strictly from the stand-
point of the Newtonian exposition in the Quadrature
of Curves, 1704.

69. James Stirling uses x and y as infinitesimals
in his Linece tertii or dints, Oxford, 1717. He draws
the infinitely small right triangle at the contact
of a curve with its asymptote, the horizontal side
being "quam minima" and equal to x, the vertical
side being y. In the appendix to this booklet of
1717, x and y are again infinitely small. In his
Methodus dijferentialis , London, 1730, there is no
direct attempt to explain fundamentals, any more
than there was in 1717, but on p. 80 he puts the
fluxion of an independent variable equal to unity,
from which we infer that a fluxion is with him now
a finite velocity.

70. For twenty-four years after Ditton no new
text appeared. In 1730 Edmund Stone, a self-
taught mathematician who had studied De PHospital,

PRINTED BOOKS, ETC., BEFORE 1734 51

sent forth a new book, the first part of which was
a translation.1

The following extract is from Stone's translation
of De PHospital's Preface, the words in the square
brackets [ ] being interpolated by Stone : —

"By means of this Analysis we compare the
infinitely small (Differences or) Parts of finite
Magnitudes, and find their Ratio's to each other ;
and hereby likewise learn the Ratio's of finite
Magnitudes, those being in reality so many infinitely
great Magnitudes, in respect of the other infinitely
small ones. This Analysis may ever be said to go
beyond the Bounds of Infinity itself; as not being
confined to infinitely small (Differences or) Parts,
but discovering the Ratio's of Differences of Differ-
ences, or of infinitely small Parts of infinitely small
Parts, and even the Ratio's of infinitely small Parts
of these again, without End. So that it not only
contains the Doctrine of Infinites, but that of an
Infinity of Infinites. It is an Analysis of this kind
that can alone lead us to the Knowledge of the true
Nature and Principles of Curves : For Curves being
no other than Polygons, having an Infinite Number
of Sides, and their Differences arising altogether
from the different Angles which their infinitely
small Sides make with each other, it is the Doctrine
of Infinites alone that must enable us to determine
the Position of these Sides, in order to get the

1 The Method of Fluxions, both Direct and Inverse. The former being
a Translation from the Celebrated Marquis De rHospitaCs Analyse
des Injinements Petit s : And the Latter Supply* d by the Translator,
E. Stone, F.R.S. London, MDCCXXX.

52 LIMITS AND FLUXIONS

Curvature formed by them ; and thence the Tangents,
Perpendiculars, Points of Inflexion and Retrogres-
sion, reflected and refracted Rays, etc., of Curves.

"Polygons circumscribed about or inscribed in
Curves, whose Number of Sides infinitely augmented
till at last they coincide with the Curves, have
always been taken for Curves themselves. ... It
was the Discovery of the Analysis of Infinites that
first pointed out the vast Extent and Fecundity of
this Principle. . . . Yet this itself is not so simple
as Dr. Barrow afterwards made it, from a close
Consideration of the Nature of Polygons, which
naturally represent to the Mind a little Triangle
consisting of a Particle of a Curve (contained
between two infinitely near Ordinates), the Differ-
ence of the correspondent Absciss's ; and this
Triangle is similar to that formed by the Ordinate,
Tangent, and Subtangent. . . . Dr. Barrow . . . also
invented a kind of Calculus suitable to the Method
(Lect. Geom., p. 80), tho' deficient. . . . The
Defect of this Method was supplied by that of
Mr. Leibnitz'z,1 [or rather the great Sir Isaac
Newton^ He began where Dr. Barrow and others
left off: His Calculus has carried him into Countries
hitherto unknown. ... I must here in justice own
(as Mr Leibnitz himself has done, in Journal des
S^avans for August 1694) that the learned Sir Isaac
Newton likewise discover'd something like the
Calculus Differentiates , as appears by his excellent

1 Acta Erudit. Lips., arm. 1684, p. 467.

2 See Conunerciuni EpistolicutH.

PRINTED BOOKS, ETC., BEFORE 1734 53

Principia, published first in the Year 1687, which
almost wholly depends upon the Use of the said
Calculus. But the Method of Mr. Leibnitz'z is
much more easy and expeditious, on account of
the Notation he uses. . . ."

In the preface of " The Translator to the Reader "
Stone points out that the work he is bringing out
' * becomes the more necessary, because there are
but two English Treatises on the Subject . . . the
one being Hay's Introduction to Mathematical
Philosophy, and the other, Ditton's Institution of
Fluxions"; the former "too prolix," the latter
"much too sparing in Examples" and "too re-
dundant " in the explanation of fluxions, so that * ' it
is next to impossible for one who has not been
conversant about Infinites to apprehend it. That
of our Author is much easier, tho less Geo-
metrical, who calls a Differential (or Fluxion) the
infinitely small Part of a Magnitude." "But," con-
tinues Stone, " I would not here be thought in any
wise to lessen the Value of Sir Isaac Newton's
Definition : When the Learner has made some
Progress, I would have him then make himself
Master of it. " Stone then proceeds to explain the
nature of fluxions, following closely Newton's
language in his Quadrature of Curves.

71. In De PHospitaPs treatise, as translated by
Stone, we read :

"The infinitely small Part whereby a variable
Quantity is continually increased or decreas'd, is
called the Fluxion of that Quantity."

54 LIMITS AND FLUXIONS

Here Stone simply writes ''fluxion" where
De 1'Hospital writes "difference," which is a
mischievous procedure, seeing that the two words
stand for things totally different. De 1'HospitaPs
wording is "La portion infiniment petite dont une
quantite variable augmente ou diminue continuelle-
ment, en est appellee la Difference." Stone also
changes from the Leibnizian to the Newtonian
notation, by writing x instead of dx. Then follow
two postulates :

"Grant that two Quantities, whose Difference
is an infinitely small Quantity, may be taken (or
used) indifferently for each other : or (which is the
same thing) that a Quantity, which is increased or
decreas'd only by an infinitely small Quantity, may
be consider'd as remaining the same.

' ' Grant that a Curve Line may be consider'd
as the Assemblage of an infinite Number of in-
finitely small right Lines : or (which is the same
thing) as a Polygon of an infinite Number of Sides,
each of an infinitely small Length, which determine
the Curvature of the Line by the Angles they make
with each other. "

De 1'Hospital's " prendre la difference" is
rendered by Stone "to find the fluxions." The
fluxion of xy is found by taking the product of x+x
and j^+jy, and neglecting xy, "because . xy is a
Quantity infinitely small, in respect of the other
Terms yx and xy. "

72. Further on in Stone's translation (p. 73) we
read :

PRINTED BOOKS, ETC., BEFORE 1734 55

"The infinitely small Part generated by the con-
tinual increasing or decreasing of the Fluxion of a
variable Quantity, is called the Fluxion of the
Fluxion of that Quantity, or second Fluxion." In
like manner he defines third Fluxion ; ' * fluxion of
the second fluxion" taking the place of "difference
de la difference seconde."

In the appendix, containing Stone's Inverse
Method of Fluxions, a fluent is defined thus :

1 ' The fluent or flowing Quantity of a given
fluxionary Expression, is that Quantity whereof
the given fluxionary Expression is the Fluxion."

Remarks

73. The earliest treatment of the new analysis
which became current in England was that of
Leibniz. The Scotsman Craig used it for over
a quarter of a century before rejecting it in favour
of fluxions. Harris, Hayes, and Stone drew their
inspiration from French writers who followed
Leibniz. A hopeless confusion arose in the use of /
the term "fluxion." Newton always took it to be
a velocity, but many writers, including Newton's
friends who prepared the Commercium Epistolicum, I
simply said "fluxion" instead of "differential,"'
thus putting a home label upon goods of foreign
manufacture. A strict follower of the Newton of
1704 was Ditton ; fluxions are taken as infinitesimals
by Fatio de Duillier, Cotes (in 1701), Harris, Hayes,
Raphson, Stirling (in 1717), and Stone.

Stone comes out strongly with the view that a

56 LIMITS AND FLUXIONS

circle is a polygon of an infinite number of sides.
He also uses the infinitesimal triangle. Hayes and
Stone have no hesitation in speaking of " fluxions
of fluxions," and ''infinitely little parts of an in-
finitely little part." No writers, unless we except
Newton (1704) and Ditton, dispense with the use of
infinitely small quantities. The dropping of such
quantities from an equation was usually permitted
without scruple.

What an opportunity did this medley of untenable
philosophical doctrine present to a close reasoner
and skilful debater like Berkeley ! [See Addenda,
p. 289.]

CHAPTER III

BERKELEY'S ANALYST (1734); CONTROVERSY
WITH JURIN AND WALTON

74. BISHOP BERKELEY'S publication of the Analyst1
is the most spectacular event of the century in the
history of British mathematics. The arguments in
the Analyst were so many bombs thrown into the
mathematical camp.

The views expressed in the Analyst are fore-
shadowed in Berkeley's Principles of Human Know-
ledge (§§ 123-134), published nearly a quarter of
a century earlier. The "Infidel mathematician,"
it is generally supposed, was Dr. Halley. Mathe-
maticians complain of the incomprehensibility of
religion, argues Berkeley, but they do so unreason-
ably, since their own science is incomprehensible.
"Our Sense is strained and puzzled with the
perception of objects extremely minute, even so
the Imagination, ... is very much strained and
puzzled to frame clear ideas of the least particles of
time, or the least increments generated therein :

1 J^he Analyst ; or, a Discourse addressed to an Infidel Mathe-
matician. Wherein it is examined whether the Object, Principles, and
Inferences of the Modern Analysis are more distinctly conceived, or
more evidently deduced, than religious Mysteries and Points of Faith.
London, 1734.

58 LIMITS AND FLUXIONS

and much more so to comprehend the moments, or
those increments of the flowing quantities in statu
nascenti, in their very first origin or beginning to
exist, before they become finite particles. And it
seems still more difficult to conceive the abstracted
velocities of such nascent imperfect entities. But
the velocities of the velocities — the second, third,
fourth, and fifth velocities, etc. — exceed, if I mistake
not, all human understanding " (Analyst, § 4). ...

75. "In the calculus differentialis . . . our
modern analysts are not content to consider only
the differences of finite quantities : they also
consider the differences of those differences, and
the differences of the differences of the first differ-
ences : and so on ad infinitum. That is, they
consider quantities infinitely less than the least
discernible quantity ; and others infinitely less than
those infinitely small ones ; and still others infinitely
less than the preceding infinitesimals, and so on
without end or limit " (§ 6).

76. "I proceed to consider the principles of this
new analysis. . . . Suppose the product or rectangle
AB increased by continual motion : and that the
momentaneous increments of the sides A and B are
a and b. When the sides A and B are deficient, or
lesser by one-half of their moments, the rectangle
was A-itfxB-i b, i.e. AB- | #B-| £A + J ab.
And as soon as the sides A and B are increased by
the other two halves of their moments, the rectangle

becomes A + J0 x B + |£ or AB + 1 # B + J £A + -J- ab.
From the latter rectangle subduct the former, and

BERKELEY'S ANALYST (1734) 59

the remaining difference will be #B + £A. There-
fore the Increment of the rectangle generated by
the entire increments a and b is #B + £A. Q.E. D.
But it is plain that the direct and true method to
obtain the moment or increment of the rectangle
AB, is to take the sides as increased by their whole
increments, and so multiply them together, A + a by
B + £, the product whereof AB + tfB + £A + #£ is the
augmented rectangle ; whence, if we subduct AB
the remainder aR + &A+a6 will be the true incre-
ment of the rectangle, . . . and this holds uni-
versally by the quantities a and b be what they
will, big or little, finite or infinitesimal, increments,
moments, or velocities " (§ 9). ... The point of
getting rid of ab cannot be obtained by legitimate
reasoning." . . .

77. "The points or mere limits of nascent lines
are undoubtedly equal, as having no more magnitude
one than another, a limit as such being no quantity.
If by a momentum you mean more than the very
initial limit, it must be either a finite quantity
or an infinitesimal. But all finite quantities are
expressly excluded from the notion of a momentum.
Therefore the momentum must be an infinitesimal.
. . . For aught I see, you can admit no quantity
as a medium between a finite quantity and nothing,
without admitting infinitesimals" (§ n).

78. Berkeley next premises the following lemma,
which figures prominently in the debates about
fluxions :

" 'If, with a view to demonstrate any proposition,

60 LIMITS AND FLUXIONS

a certain point is supposed, by virtue of which
certain other points are attained ; and such* supposed
point be itself afterwards destroyed or rejected by
a contrary supposition ; in that case, all the other
points attained thereby, and consequent thereupon,
must also be destroyed and rejected, so as from
thenceforward to be no more supposed or applied in
the demonstration.' This is so plain as to need no
proof »(§ 12).

79. Berkeley examines now the method of obtain-
ing the fluxion of xn by writing x + o in the place
of x, expanding by the binomial formula, writing
down the intrements of x and x*, which are in the
ratio of

i to nx*-l + x*-*+ etc.,

or, when the increment o is made to vanish, in the
ratio of I to nxn~^. Berkeley argues :

"But it should seem that this reasoning is
not fair or conclusive. For when it is said, let
the increments vanish, i.e. let the increments be
nothing, or let there be no increments, the former
supposition that the increments were something, or
that there were increments, is destroyed, and yet a
consequence of that supposition, i.e. an expression
got by virtue thereof, is retained. Which, by
the foregoing lemma, is a false way of reasoning.
Certainly when we suppose the increments to vanish,
we must suppose their proportions, their expres-
sions, and everything else derived from the supposi-
tion of their existence, to vanish with them (§ 13).

BERKELEY'S ANALYST (1734) 61

. . . All which seems a most inconsistent way of
arguing, and such as would not be allowed of in
Divinity (§ 14). . . . Nothing is plainer than that
no just conclusion can be directly drawn from
two inconsistent suppositions (§15). . . . It may
perhaps be said that [in the calculus differentials]
the quantity being infinitely diminished becomes
nothing, and so nothing is rejected. But, accord-
ing to the received principles, it is evident that no
geometrical quantity can by any division or sub-
division whatsoever be exhausted, or reduced to
nothing. Considering the various arts and devices
used by the great author of the fluxionary method ;
in how many lights he placeth his fluxions ; and in
what different ways he attempts to demonstrate the
same point ; one would be inclined to think, he was
himself suspicious of the justness of his own demon-
strations, and that he was not enough pleased with
any notion steadily to adhere to it" (§17). . . .

80. "And yet it should seem that, whatever;
errors [in the calculus differentialis~\ are admittec
in the premises, proportional errors ought to be
apprehended in the conclusion, be they finite o
infinitesimal : and that therefore the aKpt/3eia. o
geometry requires nothing should be neglected o
rejected. In answer to this you will perhaps sayj,
that the conclusions are accurately true, and thajt
therefore the principles and methods from whence
they are derived must be so too. But . . . th^
truth of the conclusion will not prove either the form
or the matter of a syllogism to be true" (§ 19).

62 LIMITS AND FLUXIONS

8 1. Berkeley proceeds to show that correct results
are derived from false principles by a compensation
of errors, a view advanced again later by others,
particularly by the French critic L. N. M. Carnot.
Taking y*=px, Berkeley says that the subtangent
is not ydx / dy if dy is the true increment of y
corresponding to dx ; the accurate subtangent,
obtained by similar triangles, \sydx / (dy + s\ where
z = dydy / (2y). That is, if dy is the true increment,
then in ydx I dy there is an "error of defect." But
in ydx I dy, as used in the differential calculus, the
dy is not its true value, viz. dy—pdxl(2y) —
dydy / (2y) (obtained by writing x-\-dx for x and
y-\-dy forj/, in the equation y*=-px\ but its erroneous
value, pdx / (2j/). There is here an ' ' error of
excess." "Therefore the two errors being equal
and contrary destroy each other (§ 21); . . . by
virtue of a twofold mistake you arrive, though not
at science, yet at truth." Berkeley gives other
illustrations of cases where " one error is redressed
by another."

82. "A point may be the limit of a line : a line
may be the limit of a surface : a moment may
terminate time. But how can we conceive a velocity
by help of such limits? It necessarily implies both
time and space, and cannot be conceived without
them. And if the velocities of nascent and evan-
escent quantities, i.e. abstracted from time and
space, may not be comprehended, how can we
comprehend and demonstrate their proportions ; or
consider their rationes primes and ultima? For, to

BERKELEY'S ANALYST (1734) 63
consider the proportion or ratio of things implies

o-f ~\*

that such things have magnitude ; that such their °^
magnitudes may be measured" (§31). . . .

83. l( If it be said that fluxions may be expounded
or expressed by finite lines proportional to them ;
which finite lines, as they may be distinctly con-
ceived and known and reasoned upon, so they may
be substituted for the fluxions, ... I answer that
if, in order to arrive at these finite lines proportional t
to the fluxions, there be certain steps made use of
which are obscure and inconceivable, be those
finite lines themselves ever so clearly conceived, it
must nevertheless be acknowledged that your pro-
ceeding is not clear nor your method scientific "

(§ 34).

Berkeley discusses this matter with reference to
a geometric figure, and argues that "a point there-
fore is considered as a triangle, or a triangle is
supposed to be formed in a point. Which to con-i
ceive seems quite impossible " (§ 34). . . .

84. " And what are these fluxions ? The Veloci-
ties of evanescent increments. And what are these
same evanescent increments ? They are neither
finite quantities, nor quantities infinitely small, nor 1
yet nothing. May we not call them the ghosts of
departed quantities?" (§ 35). . . .

" And if the first [fluxions] are incomprehensible,
what shall we say of the second and third fluxions,
etc.? "(§44).

"To the end that you may more clearly com-
prehend the force and design of the foregoing

64 LIMITS AND FLUXIONS

remarks . . ., I shall subjoin the following Queries"

(§ 50).

Then follow sixty-seven queries, of which the
sixteenth is a good specimen : " Qu. 16. Whether
certain maxims do not pass current among analysts
which are shocking to good sense? And whether
the common assumption, that a finite quantity
divided by nothing is infinite, be not of this
number ? "

Jurin s First Reply to Berkeley

85. A reply to Berkeley's Analyst was made by
the noted physician, James Jurin, at one time a
student in Trinity College, Cambridge, who had
imbibed Newtonian teachings from Newton himself.
Jurin wrote under the pseudonym of " Philalethes
Cantabrigiensis." The letter1 is dated April 10,

1/34.

86. Philalethes says that the charge in the
Analyst "consists of three principal points : (i) Of
Infidelity with regard to the Christian Religion.
(2) Of endeavouring to make others Infidels, and
succeeding in those endeavours by means of the
deference which is paid to their judgment, as being

1 Geometry No Friend to Infidelity : or, a Defence of Sir Isaac
Newton and the British Mathematicians, In a Letter to the Author of
the Analyst. Wherein it is examined, How far the Conduct of such
Divines as intermix the Interest of Religion with their private Disputes
and Passions, and allow neither Learning nor Reason to those they
differ from, is of Honour or Service to Christianity, or agreeable to the
Example of our Blessed Saviour and his Apostles, By Philalethes
Cantabrigiensis. Ne Deus intersit, nisi dignus vindice nodus Inciderit.
London : Printed for T. Cooper at the Globe in Ivy-Lane.
MDCCXXXIV. Price is.

BERKELEY'S ANALYST (1734) 65

presumed to be of all men the greatest masters of
reason. (3) Of error and false reasoning in their
own science."

87. The early part of Jurin's reply is given to a
discussion of the religious side. If there is no
more certainty in modern analysis, argues Jurin,
than in the Christian religion, this comparison brings
no honour to Christianity ; it is not true that
mathematicians are infidels, leading others to
infidelity. If it were true, this fact ought not in
prudence to be published. Even if it be shown
that the method of fluxions is built upon false
principles, will it follow that all other parts of
mathematics rest on inaccurate and false reasoning ?
Your attack, I surmise, is really, not so much in
the interest of Christianity, as to demonstrate your
superiority as a reasoner, by showing Newton and
Barrow, two of the greatest mathematicians, less
clear and just than you are. But because a mathe-
matician "is thought to reason well in Geometry,"
his " decisions against the Christian Religion " will
not "pass even upon weak and vulgar minds."
"Sir Isaac Newton was a greater Mathematician
than any of his contemporaries in France, . . . yet
I have not heard that the French Mathematicians
are converted to the Protestant Religion by his
authority." Your objections against Newton's
Fluxions may be ' ' reduced under three heads :
(i) Obscurity of this doctrine ; (2) False reasoning
in it by Sir Isaac Newton, and implicitly received
by his followers ; (3) Artifices and fallacies used by

66 LIMITS AND FLUXIONS

Sir Isaac Newton, to make this false reasoning pass
upon his followers." Jurin continues: "It must
be owned that this doctrine ... is not without
difficulties," but "have you not altered his ex-
pressions in such a manner, as to mislead and con-
found your readers, instead of informing them,"
thereby increasing the difficulties? "Where do
iyou find Sir Isaac Newton using such expressions
as the velocities of the velocities, the second, third and
fourth velocities, the incipient celerity of an incipient
celerity, the nascent augment of a nascent augment ? "
As to the "moment or increment of the rectangle
AB," the mathematicians take it to be
you say that the rigorous value is
"Do not they know that in estimating any finite
quantity how great soever . . ., a globe, suppose,
as big as the earth, ... or even the orb of the
fixed stars . . ., this omission shall not cause them
to deviate from the truth so much as a single pin's
head, nay not the millionth part of a pin's head ? "
The operations by fluxions are no more objection-
able than those by decimal fractions, where we take
•33333, etc., instead of J. You say that the Marquis
de 1'Hospital, in his Analyse des infiniment petits,
Prop. 2, having found the fluxion of xy to be xdy
+ydx + dxdy, drops the dxdy "without the least
ceremony. " But does he not especially require in
a postulate, "that a quantity, which is augmented
or diminished by another quantity infinitely less
than the first, may be considered as if it continued
the same, i.e. had received no such augmentation or

BERKELEY'S ANALYST (1734) 67

diminution ? " As to Newton, he takes (Principia,
lib. ii, lemma 2, cas. I ; our § 17) initially (A — \d)
(B-|£) and finally ( A + \ a)(E + J b\ thereby de-
riving tfB + £A, not as the increment of AB, but
as the increment of ( A — 1 a)(B — J b). ". . .
Rigorously speaking, the moment of the rectangle
AB is not, as you suppose, the increment of the
rectangle AB ; but it is the increment of the rect-
angle A — J a x B — \ b. " A moment may be either
an increment or a decrement ; you obtain the
increment a& -\-bA-\-ab, the decrement of AB is
aB + 6A —ab. Which of those two will you call the
moment of AB ? "I apprehend the case will stand
thus : aB-\-bA + ab + a~B + bA — ab making twice the
moment of the rectangle AB ; it follows that #B + £A
will make the single moment of the same rectangle";1
the velocity which the flowing rectangle has, is its
velocity "neither before nor after it becomes AB,
but at the very instant of time that it is AB." In
like manner with the moment of the rectangle.
Let me advise you hereafter to "first examine and
weigh every word he [Newton] uses." Lastly, 1
must observe that the moment of AB, namely
tfB + $A, and the increment of the same rectangle,
a*B-\-&A+a&, "are perfectly and exactly equal,
supposing a and b to be diminished ad infinitum"

88. As to your second instance of false reason-
ing, in Newton's book on Quadratures, apparently
that is "so truly Boeotian a blunder" that I know
not how "a Newton could be guilty of it." You

1 Jurin, op. <:*'/., p. 46.

68 LIMITS AND FLUXIONS

interpret " Evanes cant jam augmenta ilia" (our § 32),
as "let now the increments vanish, i.e. let the
increments be nothing, or let there be no incre-
ments. " But ' ' do not the words ratio ultima stare
us in the face, and plainly tell us that though there
is a last proportion of evanescent increments, yet
there can be no proportion of increments which are
nothing, of increments which do not exist?" You
grossly misinterpreted Newton.

89. As to the third head of your objections,
since Newton did not reason falsely, ' ' he had no
occasion to make use of arts and fallacies to impose
upon his followers." "Having now . . . driven
you entirely out of your intrenchments ... I
should sally out and attack you in your own."
"But as they seem rather designed for shew, than
use, ... to dazzle the imagination . . . [they]
will likewise immediately disappear like the Ghost
of a departed quantity," if you exorcise them
"with a few words out of the first section of the
Principia. " You say that the paradox, ' ' that
Mathematicians should deduce true Propositions
from false Principles " is accounted for by the fact
that one error "is compensated by another con-
trary and equal error." But the two are no errors
at all, as is evident from the fact that true results
follow when only the first operation is carried out,
so that no compensation is possible. Jurin argues
that the first supposed fallacy, without the second,
gives as the subtangent of y* = ax, the value
-r(2y)] the second supposed fallacy,

BERKELEY'S ANALYST (1734) 69

without the first, gives 2x(2y) -=- (2y + dy\ Both
these expressions are equal to 2x, "which is the
result either of two errors, or of none at all." If
you claim that ix(2y + dy)-:r(2y)> 2^r, how much
greater is it, supposing 2x— 1000 miles? Not as
much as the thousand-millionth part of an inch.
Jurin ends with a discussion of Lock on abstract
ideas.

Walton's First Reply .to Berkeley

90. Little is known about John Walton. He
was Professor of Mathematics in Dublin, and partici-
pated in this controversy. Otherwise, practically
nothing about him has been handed down.

His reply to Berkeley was published in 1735 at
Dublin.1 Berkeley attacked the method of fluxions
more particularly as given in Newton's earlier
exposition ; Walton defended the theory on the
basis of the later treatment as given by Newton
in his Quadratures Curvarum (1704), and in the
Principia, Book II.

91. Walton begins by stating that inasmuch as
the credulous may " become infected" by Berkeley's
attack on fluxions, it seems necessary to give a
short account of the nature of fluxions. "The
momentaneous Increments or Decrements of flow-

1 A Vindication of Sir Isaac Newton's Principles of Fluxions,
against the Objections contained in the Analyst. By J. Walton. —
Siquid novisti rectius istis, candidus imperti : bi non, his utere mecum.
Hor. In the fulness of his Sufficiency he shall be in Straits : Every
Hand of the Wicked shall come upon him. Job. — Dublin, Printed ;
and reprinted at London, and sold by J. Roberts in Warwick-Lane,
1735- [Price Six Pence-!

70 LIMITS AND FLUXIONS

ing Quantities, he [Newton] elsewhere calls by the
name of Moments, . . : By Moments we may
understand the nascent or evanescent Elements or
Principles of finite Magnitudes, but not Particles
of any determinate Size, or Increments actually
generated ; for all such are Quantities, themselves
generated of Moments."

92. "The magnitudes of the momentaneous
Increments or Decrements of Quantities are not
regarded in the Method of Fluxions, but their first
or last Proportions only ; that is, the Proportions
with which they begin or cease to exist." . . .
"The ultimate Ratios with which synchronal
Increments of Quantities vanish, are not the Ratios
of finite Increments, but Limits which the Ratios
of the Increments attain, by having their magni-
tudes infinitely diminish'd. . . . There are certain
determinate Limits to which all such Proportions
perpetually tend, and approach nearer than by any
assignable Difference, but never attain before the
Quantities themselves are infinitely diminish'd ;
or 'till the Instant they evanesce and become
nothing." "The Fluxions of Quantities are very
nearly as the Increments of their Fluents generated
in the least equal Particles of Time," and they
"are accurately in the first or last Proportions of
their nascent or evanescent Increments." "The
Fluxions of Quantities are only velocities. ..."
Again, ". . . to obtain the Ratios of Fluxions,
the corresponding synchronal or isochronal Incre-
ments must be lessened in infinilum. For the

BERKELEY'S ANALYST (1734) 71

Magnitudes of synchronal or isochronal Increments
must be infinitely diminished and become evan-
escent, in order to obtain their first or last Ratios,
to which Ratios the Ratios of their corresponding
Fluxions are equal. " The moment of the rectangle
AB is A£+Btf, for consider Ab+¥>a + ab and A£ +
B#, "under a constant Diminution of the Incre-
ments a and b . . . [they] constantly tend to an
Equality . . . [and] they become equal, and their
Ratio becomes a Ratio of Equality. ..." Hence
A£+B<z + tf^ "is not the Moment or Fluxion of
the Rectangle AB, except in the very Instant
when it begins or ceases to exist." Here fluxions
^appear to be no longer velocities (finite magnitudes)
but moments. Walton next quotes a Latin passage
from the Quadratures Cutvarum. He says that
Berkeley seems "to have been deceived by an
Opinion that there can be no first or last Ratios
of mathematical Quantities," but Walton insists
that if quantities are generated together, or if they
vanish together, they will do so "under certain
Ratios, which are their first or last Ratios."
Walton claims that Berkeley's lemma "is in no
Way pertinent to the Case for which it was in-
tended " ; he explains the Newtonian process of
finding the fluxion of xn, supposing x to increase
uniformly, and points out that this is done without
rejecting quantities "on account of their exceeding
smallness." Commenting on Berkeley's contention
that "no geometric Quantity, by being infinitely
diminished, can ever be exhausted or become

72 LIMITS AND FLUXIONS

nothing," Walton states that the fluxional calculus
assumes that "Quantities can be generated by
Motion . . . and consequently they may also by
Motion be destroy'd. "

93. Walton's Vindication follows Newton's ex-
position closely ; Berkeley's claim that Walton
followed in Jurin's track and borrowed from him,
is, I believe, incorrect. Take the vital question of
rejecting infinitesimals : Jurin claims that, being
so very small, they do not appreciably affect the
result ; Walton takes the stand that there is no
rejection whatever of infinitesimals. The main
criticism to be passed on Walton's first essay con-
sists, in our judgment, in a failure to meet
Berkeley's objections squarely and convincingly.

Berkeley's Reply to Jurin and Walton

94. Jurin's and Walton's articles were answered
by Berkeley in a publication entitled, A Defence of
Free- Thinking in Mathematics. 1

Berkeley restates the purpose he had in writing
the Analyst-. "Now, if it be shewn that fluxions
are really most incomprehensible mysteries, and
that those who believe them to be clear and scien-
tific do entertain an implicit faith in the author of
that method : will not this furnish a fair argumen-
tum ad hominem against men who reject that very
thing in religion which they admit in human learn-

1 A Defence of Free-Thinking ni Mathematics. In Answer to a
Pamphlet of Philalethes Cantabrigiensis. . . . Also an Appendix
concerning Mr. Waltorfs Vindication. . . . By the Author of " The
Minute Philosopher," Dublin, 1735.

BERKELEY'S ANALYST (1734) 73

ing?1 (§ 3) . . . I say that an infidel, who believes
the doctrine of fluxions, acts a very inconsistent
part in pretending to reject the Christian religion
— because he cannot believe what he doth not
comprehend" (§ 7). ...

1 Berkeley is not the only one who invoked the aid of the Doctrine
of Fluxions in theological discussion. In a criticism (A Revieiv of the
Fiery Eruption, etc., London, 1752, p. 128) of Bishop William
Warburton's Julian, concerning earthquakes and fiery eruptions,
which, Warburton argued, defeated Julian's attempt to rebuild the
temple at Jerusalem, it is stated that a connection (needed in the
argument) was established between the preservation of Christianity
and the destruction of Judaism by the following clever procedure : —

"The great modern Father of the mathematics had invented a new
and curious way of improving that science by a fiction ; according to
which quantities are supposed to be generated by the continual flux or
motion of others. In the application of this method it became neces-
sary to consider these quantities, sometimes in a nascent, and at other
times in an evanescent state, by which ingenious contrivance they
could be made either continually to tend to and at last absolutely to
become nothing, or vice versa, according to the intention and occasions
of the Artist. Now by extending this noble invention to the two
religions, it evidently appeared, that, from the time of the first coming
of Christ, Judaism entered into its evanescent state, as on the other
hand Christianity did into a nascent state, by which means both being
put into a proper flux, one was seen continually decaying, and the other
continually improving, till at last by the destruction of the Temple
Judaism actually vanished and became nothing, and the Christian
religion then bursted out a perfectly generated Entity. ... As the
great author of the mathematical method of fluxions had for very good
reasons studiously avoided giving any definition of the precise magni-
tude of those moments, by whose help he discovers the exact magnitude
of the generated quantities, so our Author [Warburton] by the same
rule of application, and under the influence of the same authority, was
fairly excused from defining that precise degree of perfection and
imperfection in which the two religions subsisted, during the respective
evanescent and nascent state of each, by the help of which he discovered
the precise time when Judaism was perfectly abolished, and Christianity
perfectly established. But we may well suppose, that the most alluring
charm in this extraordinary piece of ingenuity, was the creating of a
new character by it : For questionless he may now be justly stiled the
great founder and inventor of the ftuxionary method of theology. . . .
This fancy of a necessary connexion between the Temple-edifice, and
the being of Christianity, . . . this pretended Christianity which is of
such an unsubstantial nature, that it must necessarily vanish at the
restoration of the Temple, can be nothing else but a mere Ghost, . . .
evidently the Ghost of departed Judaism."

74 LIMITS AND FLUXIONS

95. "I have said (and I venture still to say) that
a fluxion is incomprehensible : that second, third,
and fourth fluxions are yet more incomprehensible :
that it is not possible to conceive a simple infini-
tesimal : that it is yet less possible to conceive an
infinitesimal of an infinitesimal, and so onward.
What have you to say in answer to this ? Do you
attempt to clear up the notion of a fluxion or a
difference? Nothing like it" (§ 17).

96. Berkeley quotes from Newton's Principia
and Quadrature of Curves, and then asks, "Is it
not plain that if a fluxion be a velocity, then the
fluxion of a fluxion may, agreeably thereunto, be
called the velocity of a velocity ? In like manner,
if by a fluxion is meant a nascent augment, will it
not then follow that the fluxion of a fluxion or
second fluxion is the nascent augment of a nascent
augment?" (| 23).

97. "I had observed that the great author had
proceeded illegitimately, in obtaining the fluxion
or moment of the rectangle of two flowing quan-
tities. ... In answer to this you allege that the
error arising from the omission ... is so small
that it is insignificant (§ 24). ... If you mean
to defend the reasonableness and use of approxi-
mations ... I have nothing to say. . . . That
the method of fluxions is supposed accurate in
geometrical rigour is manifest to whoever considers
what the great author writes about it ... In
rebus mathernaticis errores quam minimi non sunt
contemnendi" (§ 25 ; our § 30).

BERKELEY'S ANAL YST (1734) 75

98. Berkeley justifies his use of the expression
" increment of a rectangle " by quoting from Newton
(our § 17), " rectanguli incrementum 0B + M.."

"You say 'you do not consider AB as lying at
either extremity of the moment, but as extended
to the middle of it ; as having acquired the one
half of the moment, and as being about to acquire
the other ; or, as having lost one half of it, and
being about to lose the other.' Now, in the name
of truth, I entreat you to tell what this moment
is, ... Is it a finite quantity, or an infinitesimal,
or a mere limit, or nothing at all ? . . . If you
take it in either of the two former senses, you con-
tradict Sir Isaac Newton. And, if you take it in
either of the latter, you contradict common sense ;
it being plain that what hath no magnitude, or is
no quantity, cannot be divided " (§ 30).

"... You observe that the moment of the
rectangle determined by Sir Isaac Newton, and the
increment of the rectangle determined by me are
perfectly and exactly equal, supposing a and b to
be diminished ad infinitum : and, for proof of this,
you refer to the first lemma of the first section of
the first book of Sir Isaac's Principles. I answer
that if a and b are real quantities, then ab is some-
thing, and consequently makes a real difference :
but if they are nothing, then the rectangles
whereof they are coefficients become nothing like-
wise : and consequently the momentum or incre-
mentum, whether Sir Isaac's or mine, are in that
case nothing at all. As for the above-mentioned

76 LIMITS AND FLUXIONS

lemma, . . . however that way of reasoning may
do in the method of exhaustions, where quantities
less than assignable are regarded as nothing ; yet,
for a fluxionist writing about momentums, to argue
that quantities must be equal because they have no
assignable difference, seems the most injudicious
step that could be taken ; . . . for, it will thence
follow that all homogeneous momentums are equal,
and consequently the velocities, mutations, or
fluxions, proportional thereto, are all likewise
equal" (§ 32).

99. As regards Newton's evane scant jam augmenta
ilia (our § 32), Berkeley argues that it means either
"let the increments vanish," or else "let them
become infinitely small," but the latter "is not Sir
Isaac's sense," since on the very same page in the
Introduction to the Quadrature of Curves he says
that there is no need of considering infinitely small
figures. Taking advantage of the fact that the
Newton of the Principia (1687) differed from the
Newton of the Quadratura Curvarum (1704), Berke-
ley broke out into the following philippic: "You
Sir, with the bright eyes, be pleased to tell me,
whether Sir Isaac's momentum be a finite quantity,
or an infinitesimal, or a mere limit ? If you say a
finite quantity ; be pleased to reconcile this with
what he saith in the scholium of the second lemma
of the first section of the first book of his Principles
(our § 12): Cave intelligas quantitates magniiudine
determinatas , sed cogita semper diminuendos sine
limite. If you say, an infinitesimal ; reconcile this

BERKELEY'S ANALYST (1734) 77

with what is said in his Introduction to the Quadra-
tures (our § 33) : Volui ostendere quod in methodo
fluxionum non opus sit figuras infinite parvas in
geometriam introducere. If you should say, it is a
mere limit ; be pleased to reconcile this with what
we find in the first case of the second lemma in the
second book of his Principles (our § 17): Ubi de
lateribus A et B deerant momentorum dimidia, etc.—
where the moments are supposed to be divided. I
should be very glad a person of such a luminous
intellect would be so good as to explain whether
by fluxions we are to understand the nascent or
evanescent quantities themselves, or their motions,
or their velocities, or simply their proportions . . .
that you would then condescend to explain the
doctrine of the second, third, and fourth fluxions,
and show it to be consistent with common sense if
you can" (§ 36).

100. In an appendix to the Defence of Free-Think-
ing in Mathematics, Berkeley replies to Walton,
stating that the issues raised by him had been
previously raised by "the other," that he delivered
a technical discourse without elucidating anything,
that his scholars have a right to be informed as to
the meaning of fluxions and should therefore ask
him "the following questions." Then follow many
questions, of which we give a few :

"Let them ask him — Whether he can conceive
velocity without motion, or motion without ex-
tension, or extension without magnitude ? . . .
Whether nothing be not the product of nothing

78 LIMITS AND FLUXIONS

multiplied by something ; and, if so, ... when ab
is nothing, whether A&+Ba be not also nothing?
i.e. whether the momentum of AB be not nothing?
Let him then be asked, what his momentums are
good for, when they are thus brought to nothing ?
/ . . . I wish he were asked to explain the differ-
ence between a magnitude infinitely small and a
magnitude infinitely diminished. . . . Let him be
farther asked, how he dares to explain the method
of Fluxions, by the Ratio of magnitudes infinitely
diminished, when Sir Issac Newton hath expressly
excluded all consideration of quantities infinitely
small? If this able vindicator should say that
quantities infinitely diminished are nothing at all,
and consequently that, according to him, the first
and last Ratio's are proportions between nothings,
let him be desired to make sense of this. ... If
he should say the ultimate proportions are the
Ratio's of mere limits, then let him be asked how
the limits of lines can be proportioned or divided?"

Walton's Second Reply to Berkeley

101. In a second reply1 to Berkeley, Walton
states that in the Appendix to the Defence, Berkeley
"has composed a Catechism which he recommends
to my Scholars " and which Walton quotes. I am
first to be asked, " Whether I can conceive Velocity
without Motion, or Motion without Extension. . . .

1 J. Walton, Catechism of the Author of the Minute Philosopher
Fully answered. Printed at Dublin. Reprinted at London, and sold
by J. Roberts, 1735. It is a pamphlet of 30 pages.

BERKELEY'S ANALYST (1734) 79

I answer, I can conceive Velocity and Motion in a
Point of Space ; that is, without any assignable
Length or Extension described by it ... for ...
if a cause acts continually upon a given Thing . . .
there must be a continual Increase of its Velocity :
the Velocity cannot be the same in any two
different Points," as in the case of falling bodies.
Referring to A£-fB<z, Walton continues : "I agree
with him that nothing is the Product of nothing
multipl'd by something ; but must know what he
means by the vanishing of the Gnomon l and Sum
of the two Rectangles . . . before I give him a
direct Answer. If by vanishing he means that
they vanish and become nothing as Areas, I grant
they do ; but absolutely deny, upon such an Evan-
escence of the Gnomon and Sum of the two
Rectangles by the moving back of the Sides of the
Gnomon till they come to coincide with those of
the Rectangle, that nothing remains. For there
still remain the moving Sides, which are now
become the Sides of the Rectangle, . . . the
Motion of the Gnomon is the same with the Sum of
the Motions of the Two Rectangles, when they
evanesce, and are converted into the two Sides of
the Rectangle AB. If a point moves forward to
generate a Line, and afterwards the same Point
moves back again to destroy the Line with the very
same Degrees of Velocity, in all Parts of the Line

1 If a parallelogram is extended in length and breadth and if the
original parallelogram be removed, the remaining figure is called the
gnomon.

8o LIMITS AND FLUXIONS

which it had in those Parts when moving forward
to generate it ; in the Instant the Line vanishes, as
a Length . . . the generating point will remain,
together with the Velocity it had at the very
Beginning of its Motion. And the Case is the
very same with respect to the Rectangle increas-
ing by the Motion of its Sides." This point is
elaborated with great fullness. After some illustra-
tions, Walton exclaims: "This is a full and clear
Answer to this part of the catechism, and shows
that its Author has been greatly mistaken in
supposing that I explained the Doctrine of Fluxions
by the Ratio of Magnitudes infinitely diminished, or
by Proportions between nothings. ... I do not
wonder that this Author should have no clear Ideas
or Conceptions of second, third or fourth Fluxions,
when he has no clear Conceptions of the common
Principles of Motion, nor of the first and last Ratios
of the isochronal Increments of Quantities generated
and destroyed by Motion. ... In order to prevent
my being Catechised any more by this Author,"
Walton makes a confession "of some Part of my
Faith in Religion."

Juriris Second Reply to Berkeley

102. Jurin brought out a second publication,1 of
112 pages, which was in reply to Berkeley's Defence
of Free-Thinking. Passing by unimportant pre-
liminaries, we come to Jurin's definitions of " flow-

1 The Minute Mathematician : or. The Free- Thinker no Just-
Thinker. By Philalethes Cantabrigiensis. London, 1735.

BERKELEY'S ANALYST (1734) 81

ing quantity," " fluxion " ("the velocity with which
a flowing quantity increases or decreases "), "incre-
ment," "nascent increment" ("an increment just
beginning to exist from nothing . . . but not yet
arrived at any assignable magnitude how small so-
ever"), "evanescent increment" (similarly defined).
He then endeavours to prove the proposition :
' ' The Fluxions, or Velocities of flowing quantities
. . . are exactly in the first proportion of the
nascent increments, or in the last proportion of the
evanescent increments." He insists that "the first
ratio of the nascent increments must be the same,
whether the velocities be uniform or variable " ;
hence, "the nascent increments must be exactly
as the velocities with which they begin to be
generated." In further explanation, Jurin says that,
according to Newton, nascent increments are "less
than any finite magnitude," "their magnitude
cannot be assigned or determined," "the proportion
between them . . . being all that is requisite in
his Method." In further explanation of the pro-
portion of evanescent increments he says, it "is
not their proportion before they vanish," "nor is it
their proportion after they have vanished," "but it
is their proportion at the instant that they vanish."
I Jurin then states that Berkeley has "taken as
much pains as ... any man living, except a late
Philosopher of our University, to make nonsense
of Sir Isaac Newton's principles." There is no
occurrence in Newton's writings of "velocity with-
out motion," "motion without extension," which

82 LIMITS AND FLUXIONS

Berkeley pretends to derive from them. Jurin
succeeds, we think, in establishing the contention
that there is no greater difficulty in explaining the
second or third fluxion, than there is in explaining
the first. " The second fluxion is the velocity with
which the first fluxion increases." Jurin confesses
that his statement in his first reply to Berkeley, to
the effect that certain errors were of "no significance
in practice," was intended for popular consumption,
for men such as one meets in London.

103. "One of them, indeed, could make nothing
of what I had said about the length of a subtangent,
or the magnitude of the orb of the fixed stars ; but
was fully satisfied by the information given him by
one of his acquaintance to the following effect. The
Author of the Minute Philosopher has found out that,
if Sir Isaac Newton were to measure the height of
St. Paul's Church by Fluxions, he would be out about
three quarters of a hair's breadth : But yonder is one
Philalethes at Cambridge, who pretends that Sir
Isaac would not be out above the tenth part of hair's
breadth. Hearing this, and that two books had
been written in this controversy, the honest gentle-
man flew into a great passion, and after muttering
something to himself about some body's being over-
paid, he went on making reflections, which I don't
care to repeat, as not being much for your honour
or mine."

104. Jurin thereupon takes up the rectangle
AB. The terms "moment" and "increment"
are involved in the discussion of it. Jurin de-

BERKELEY'S ANALYST (1734) 83

clares : "I absolutely and fully agree with you
that the incrementum in the conclusion is the
momentum in the Lemma, " that ' * the momentum
in the Lemma" is "the momentum of the rectangle
AB." Further, Jurin says, "the incrementum in
the conclusion is manifest!)^ the excess of the
rectangle A + \ a X B + \ b> above the rectangle
A — JtfxB — \b, i.e. the increment of the rectangle
A — \ a x B — \ b. Therefore we are agreed that the
moment of the rectangle AB is the increment of
the rectangle A — | # x B — \b. Consequently you
were mistaken in supposing that the moment of the
rectangle AB was the increment of the same rectangle
AB. . . . The moment AB is neither the increment
nor the decrement of AB, " for if it really was the
increment of AB, and also its decrement, we would
have A<£ + Ea + ab = Ab + Ba — ab, i. e. 2ab — o. Hence
the rectangle ab "is by his Own confession equal to
nothing." Jurin concludes that the fluxion of AB
is not the velocity with which the increment or
decrement of AB is generated, but the "middle
arithmetical proportional between these two velo-
cities," this being "in like manner as I had
before supposed an arithmetical mean between the
increment and decrement of AB, whicK mean is the
moment of AB." Berkeley had considered four
definitions of a moment, that of a finite quantity,
or an infinitesimal or a mere limit, or nothing at
all ; and he had found each either to contradict
Newton or to contradict common sense. Jurin does
not accept "any one of those senses." A moment,

84 LIMITS AND FLUXIONS

says Jurin, is defined by Newton as "nascent
increment," its magnitude is " utterly unassignable."
Jurin continues :

105. " You seem much at a loss to conceive how
a nascent increment, a quantity just "beginning to
exist, but not yet arrived to any assignable
magnitude, can be divided or distinguished into
two equal parts. Now to me there appears no
more difficulty in conceiving this, than in appre-
hending how any finite quantity is divided or dis-
tinguished into halves. For nascent quantities may
bear all imaginable proportions to one another, as
well as finite quantities."

106. Near the close Jurin enters upon the dis-
cussion of Berkeley's Lemma, given in the Analyst :
"If one supposition be made, and be afterwards
destroy'd by a contrary supposition, then everything
that followed from the first supposition, is destroyed
with it." Not so, says Jurin, when the supposition
and its contradiction are made at different times.
"Let us imagine yourself and me to be debating
this matter, in an open field, ... a sudden violent
rain falls . . . we are all wet to the skin ... it
clears up ... you endeavour to persuade me I
am not wet.' The shower, you say, is vanished
and gone, and consequently your . . . wetness
. . . must have vanished with it." You say that
your explanation of the correctness of results as
due to a compensation of errors, was intended by
you to apply, not to Newton, but to Marquis de
1'Hospital ; your statements were such that not I

BERKELEY'S ANALYST (1734) 85

alone, but Mr. Walton as well, inferred that you
were charging Newton with committing double
errors. The rest of Jurin's ill-arranged article is
given either to a renewed and fuller elucidation of his
previous contentions or to poetical outbursts. Sure
of the soundness of his exposition, he exclaims, " I
meet with nothing in my way but the Ghosts of
departed difficulties and objections."

Berkeley's Second Reply to Walton

107. Walton's Catechism . . . fully Answered 'was
followed by Berkeley's Reasons for not replying to
Mr. Walton's Full Answer, 1735. This last reply
has been called "a combination of reasoning and
sarcasm," in which " he affects to treat his opponent
as a disguised convert." Says Berkeley : " He
seems at bottom a facetious man, who, under the
colour of an opponent, writes on my side of the
question, and really believes no more than I do of
Sir Isaac Newton's doctrine about fluxions, which
he exposes, contradicts, and confutes, with great
skill and humour, under the mask of a grave vindica-
tion." Berkeley objects to Walton's motion and
velocity "in a point" of space; " consider the
reasoning : The same velocity cannot be in two
points of space ; therefore velocity can be in a point
of space. ... I can as easily conceive Mr. Walton
should walk without stirring, as I can his idea of
motion without space. " 1 Newton calls absolute

1 Walton is not consistent in bis use of the term "motion." In some
passages it means translation ; in others it means velocity, or else both

86 LIMITS AND FLUXIONS

motion " a translation from absolute place to absolute
place,"1 and relative motion, "from one relative
place to another. Mr. Walton's is plainly neither
of these sorts of motion " ; hence, he argues against
Newton. "When ab is nothing, that is, when a
and b are nothing, he denies that A£+ Ea is nothing.
This is one of the inconsistencies which I leave the
reader to reconcile." In his Vindication he holds
that, "to obtain the last ratio of synchronal incre-
ments, the magnitude of those increments must be
infinitely diminished " ; in his Catechism . . . fully
Answered "he chargeth me as greatly mistaken in
supposing that he explained the doctrine of fluxions
by the ratio of magnitudes infinitely diminished."2
In his Catechism . . . fully Answered "he tells us
that ' fluxions are measured by the first and last
proportion of isochronal increments generated or
destroyed by motion.' A little later he says, these
ratios subsist when the isochronal increments have
no magnitude." Can "isochronal increments sub-
sist when they have no magnitude " ? Berkeley

translation and velocity, as when he says, "... isochronal increments
must be made to vanish by a Retroversion of the Motion before we can
obtain the Motions with which they vanish, or begin to be generated ;
that is, before we can obtain the Fluxions of the Quantities, the Name
given by Sir Isaac Newton to those Motions." J. Walton, Catechism
. . . fully Answered, pp. 18, 19.

1 I. Newton, Principia, Definitions, Scholium, def. viii.

2 What Walton actually wrote was, that Berkeley had been mistaken
in supposing that he explained fluxions " by the Ratios of Magnitiides
infinitely diminished) or by Proportions between nothings" Three
pages earlier Walton had denied that Newton and he measured fluxions
" by the Proportions of Magnitudes infinitely small." Evidently Walton
meant to exclude the " infinitely small," but used ' ' magnitudes infinitely
diminished" at one time as magnitudes "infinitely small," and at
another time as signifying something else, namely, "increments" that
"vanish."

BERKELEY'S ANALYST (1734) 87

then quotes from his own Analyst: "As it is im-
possible to conceive velocity without time or space,
without either finite length or finite duration, it
must seem above the power of man to comprehend
even the first fluxions." In the endeavour to
explain this matter, Walton's skill has been "vain
and impertinent."

The Second Edition of Walton's Second Reply
1 08. Walton begins 1 by explaining what Newton
means by Velocity. It is "the ratio of the
Quantity of Motion to the Quantity of Matter in
• the body " ; that is, if V is the velocity, M the
quantity of motion, F the force generating the
motion, D the density, B the bulk or magnitude,
W the weight, then " V is M / Q, and is as F / W,
or as F / DB," for, " the Quantity of Motion is the
Quantity of Matter and Velocity taken together ;
that is, M is QV " (p. 35). "The Author [Berkeley]
therefore has been grossly mistaken in asserting
that Velocity necessarily implies both Time and
Space, and cannot be conceived without them. —
And that there is no Measure of Velocity except
Time and Space." It appears that "a body in
Motion, will have a Velocity inherent in itself
during the Whole Time of its Motion : and conse-
quently there must be a Velocity where-ever the
Body is, exclusive of Time and Space ... its

1 The Catechism of the Author of the Minute Philosopher fully
answered. The Second Edition. With an Appendix, in 'Answer to the
Reasons for not replying to Mr. Walton's Fidl Answer. By J. Walton
. . . Dublin : Printed by S. Powell, for William Smith at the Hercules,
Bookseller : in Dame-Street, 1735.

88 LIMITS AND FLUXIONS

[a point's] Velocity will exist in a Point, and
successively will exist in every Point of Space
through which the Point moves " (p. 37). Berkeley
thinks that "from the generated Velocity not
being the same in any two different Points of the
described Space it will not follow that Velocity
can exist in a Point of Space. But in this he is
mistaken. For the continual Action of a Moving
Force necessarily preserves a continual Velocity ;
and if the generated Velocity be not the same in
any two different Points of the described Space, a
Velocity must of Consequence exist in every Point
of that Space " (p. 38). This account of velocity
"is agreeable to Sir Isaac Newton's Notion of
Velocity ; who constantly excludes described Space
from his Idea of that Term." Motion being
measured by QV, "the continual translation of a
Body therefore into a new Place is, ... an Effect
of this Tendency forward in the Body, and not the
Tendency itself ; consequently Space described is
an Effect of Velocity, and not Velocity itself"
(p. 47). On the question of first and last ratios it
cannot be said that Walton here throws new light.
He insists that he explained fluxions not "by the
Ratio of Magnitudes infinitely diminish'd, but by
the first and last Ratios of Increments generated or
destroyed in equal times : that is, by the Ratios of
the Velocities with which those Increments begin
or cease to exist" (p. 53). To Berkeley's charge
that Walton "supposed two Points to exist at the
same Time in one Point, and to be moved different

BERKELEY'S ANALYST (1734) 89

Ways without stirring from that Point," Walton
replies that there is no difficulty in supposing two
points existing in a given place each having its own
velocity, but he never said that they can go
in different directions "without stirring from the
Point." Berkeley, in his remarks about the fourth
fluxion of a cube, did not observe all the conditions
which he [Walton] had imposed. " He [Berkeley]
intreats me to explain whether Sir Isaac's Momentum
be a finite Quantity, or an Infinitesimal, or a mere
Limit. I tell him, that Sir Isaac's Momentum is a
finite quantity ; it is a Product contained under the ;
moving Quantity and its Velocity, or under the |
moving Quantity and first Ratio of that Space
described by it in a given Particle of Time." Since
both these factors are finite, the product is finite
(p. 62). "By Moments therefore he is not to
understand generated Increments of Fluents, but
certain finite Products or Quantities of very different
Nature from generated Increments, expressing only
the Motions with which those Increments begin or
cease to exist " (p. 63).

Remarks

109. Berkeley's Analyst must be acknowledged
to be a very able production, which marks a turning-
point in the history of mathematical thought in
Great Britain.

His contention that no geometrical quantity can
be exhausted by division x is in consonance with
1 See our § 79.

90 LIMITS AND FLUXIONS

the claim made by Zeno in his " dichotomy," and
the claim that the actual infinite cannot be realised/"
The modern reader may not agree with Berkeley
on this point, nor in the claim that second or
third fluxions are more mysterious than the first
fluxion. Nevertheless, a reader of Berkeley feels
that he spoke in the Analyst with perfect sincerity.
Interesting is De Morgan's comment : x ' * Dishonesty
must never be insinuated of Berkeley. But the
Analyst was intentionally a publication involving
the principle of Dr. Whately's argument against the
existence of Buonaparte ; and Berkeley was strictly
to take what he found. The Analyst is a tract
which could not have been written except by a
person who knew how to answer it. But it is
singular that Berkeley, though he makes his
fictitious character nearly as clear as afterwards
did Whately, has generally been treated as a
real opponent of fluxions. Let us hope that the
arch Archbishop will fare better than the arch
Bishop."

no. Sir William Rowan Hamilton once wrote
De Morgan : ' ' On the whole, I think that Berkeley
persuaded himself that he was in earnest against
Fluxions, especially of orders higher than the first,
as well as against matter." To this De Morgan
replied : " I have no doubt Berkeley knew that the
fluxions were sound enough. " 2

1 A. De Morgan, Philosophical Magazine, 4 S., vol. iv, 1852, p. 329,
note.

2 Life of Sir William Rowan Hamilton, by R. P. Graves, vol. iii,
1889, p. 581.

BERKELEY'S ANALYST (1734) 91

Hi. One is not so easily convinced of the ability
and sincerity of Jurin. That at first he should
argue that quantities may be dropped because small,
and afterwards admit that this argument was in-
tended for popular consumption, is not reassuring.1
That he should fail to see the soundness of Berke-
ley's criticism of Newton's proof ( A + J a)(B + J b)
— (A — J #)(B — | b) for the increment of AB is
somewhat surprising, even if it must be admitted
that neither Walton nor any other eighteenth-
century mathematician appears to have seen and
admitted the defect. In this connection we quote
from a letter which Hamilton wrote De Morgan
in 1862 when Hamilton was seeing his Elements of
Quaternions through the press : 2

"When your letter arrived this morning, I was
deep in Berkeley's ' Defence of Freethinking in
Mathematics ';...! think there is more than
mere plausibility in the Bishop's criticisms on the
remarks attached to the Second Lemma of the
Second Book of the Principia ; and that it is very
difficult to understand the logic by which Newton
proposes to prove, that the momentum (as he calls
it) of the rectangle (or product) AB is equal to
tfB-f-£A, if the momenta of the sides (or factors)
A and B be denoted by a and b. His mode of
getting rid of ab appeared to me long ago (I must
confess it) to involve so much of artifice, as to

1 See our §§ 97, 102, 103.

2 Life of Sir William Rowan Hamilton , by R. P. Graves, vol. iiij
p. 569.

92 LIMITS AND FLUXIONS

deserve to be called sophistical; although I should
not like to say so publicly. He subtracts, you
know, (A-J«)(B-J£) from (A + i*)(B + J£);
whereby, of course, ab disappears in the result.
But by what right, or what reason other than to
give an unreal air of simplicity to the calculation,
does he prepare the products thus ? Might it not be
argued similarly that the difference,

was the moment of A3 ; and is it not a sufficient
indication that the mode of procedure adopted is not
the fit one for the subject, that it quite masks the
notion of a limit ; or rather has the appearance of
treating that notion as foreign and irrelevant, not-
withstanding all that had been said so well before,
I in the First Section of the First Book ? Newton
does not seem to have cared for being very consis-
; tent in his philosophy , if he could anyway get hold
\( of truth, or what he considered to be such. ..."

We give also Hermann Weissenborn's objec-
tion x to Newton's procedure of taking half of
the increments a and b ; with equal justice
one might take, says he, (A + f a)(B + f b) —
(A — J a)(B — | b\ and the result would then be

1 1 2. Walton's two (or three) articles do not
seem to have been read much. They are seldom
mentioned. The pamphlets are now rare. Pro-

1 H. Weissenborn, Principien der hoheren Analysis in ihrer Ent-
wickelungvon Leibniz bis auf Lagrange, Halle, 1856, p. 42.

BERKELEY'S ANALYST (1734) 93

fessor G. A. Gibson had not seen them when he
wrote on the Analyst controversy.1 Walton seemed
to have a good intuitive grasp of fluxions, but
lacked deep philosophic insight. He showed him-
self inexperienced in the conduct of controversies, /
and did not know how to protect himself against
attack from a skilful adversary.

113. It is worthy of notice that Walton2 ex-
pressed himself on the nature of limits, by claiming
that the limit was reached. As to the nature
of " variable velocity," it is interesting to see that
Berkeley realised the difficulty of the concept, and
probably saw that there was no variable velocity as
a physical fact, while Walton was compelled to take
refuge in less primitive mechanical concepts in order
to uphold his side of the argument.3 Unjustifiable
is Walton's identification of Newton's "moment"
with " momentum" of mechanics.

114. Berkeley's Lemma4 was rejected by Jurin
and Walton. We shall see that it found no recog-
nition from mathematicians in England during the
eighteenth century, but was openly and repeatedly
accepted as valid in its application to limits, by
Woodhouse at the beginning of the nineteenth
century. The Newtonian derivation of the fluxion
of xn (see our §§ 32, 41), accomplished by dividing
both o and (x-\-6)n — xn by the finite increment 0,
and then putting o equal to zero in the quotient, is

1 G. A. Gibson, " The Analyst Controversy," in Proceedings of the
Edinburgh Math. Soc., vol. xvii, 1899, p. 18.

2 See our § 92. 3 See our § 108. 4 See our §§ 78, 92, 106.

94 LIMITS AND FLUXIONS

certainly open to the logical objection raised by
Berkeley. Eighteenth-century mathematicians did
not attach due importance to this point.

115. The existence of infinitesimals (infinitely
small quantities) was denied by Berkeley, but, it
would seem, not denied by Jurin and Walton. All
three finally abjured the philosophy which permits
their being dropped because so small. It is well
known that many mathematicians of prominence
have believed in the reality of such quantities.
From Leibniz to Lagrange all Continental writers
of note used them. Lagrange headed a small
school that was opposed to them, when he pub-
lished his Fonctions analytiques. There followed
a reaction against Lagrange. De Morgan once
remarked: "Duhamel, Navier, Cournot, are pure
infinitesimalists. Some of them say an infinitely
small quantity is one which may be made as small
as you please. This is an evasion ; but they do not
mean that dx is finite. . . . By-the-way, Poisson
was a believer in the reality of infinitely small
quantities — as I am."1

"... For myself, I am now fixed in the faith
of the subjective reality of infinitesimal quantity. But
what an infinitely small quantity is, I know no
more than I know what a straight line is ; but I
know it is ; and there I stop short. But I do not
believe in objectively realised infinitesimals. "

1 Life of Sir William Rowan Hamilton^ by Robert P. Graves,
vol. iii, pp. 572, 580. Consult also De Morgan's article, "On
Infinity ; and on the Sign of Equality," in 7rans. of the Cambridge
Phil. Society, vol. xi, Cambridge, 1871 [read May 1 6, 1864].

BERKELEY'S ANALYST (1734) 95

1 1 6. We must not neglect to express our appre-
ciation of the fact that Berkeley withdrew from the
controversy after he had said all that he had to say
on his subject. Some of the debates that came
later were almost interminable, because the par-
ticipants continued writing even after they had
nothing more to say.

CHAPTER IV

JURIN'S CONTROVERSY WITH ROBINS AND
PEMBERTON

Robins 's ( ( Discourse, " and Review of it

117. Benjamin Robins was a native of Bath and
a self - educated mathematician of considerable
reputation.

The debate carried on by Bishop Berkeley with
Jurin and Walton induced Benjamin Robins to
issue a publication, entitled, A Discourse Concerning
the Nature and Certainty of Sir Isaac Newton's
Methods of Fluxions, and of Prime and Ultimate
Ratios, I/35.1 Evidently Robins felt that Berkeley's
attacks should be met, and that Jurin was not the
man to defend Newton satisfactorily. Robins was
a man of mathematical power ; his exposition is
regarded by Professor G. A. Gibson as very able,
and far superior to that of Jurin.2 Without naming
either Berkeley or Jurin, and without referring to
their articles, Robins proceeds to his task. The
whole foundation of the doctrine of fluxions is

1 This paper is republished, along with subsequent articles on the
same subject, in the Mathematical Tracts of the late Benjamin Robins,
Esq. ... in two volumes, edited by James Wilson, M.D. London,
1761, vol. ii, pp. 1-77.

2 G. A. Gibson, loc. fit., pp. 22-25.

JURIN v.- ROBINS AND PEMBERTONi 97 )

based by Robins upon the following two defini-
tions and certain general propositions annexed to
them :

1. Definition : " . . . we . . . define an ulti-
mate magnitude to be the limit, to which a varying
magnitude can approach within any degree of
nearness whatever, though it can never be made
absolutely equal to it."

Here for the first time is the stand taken openly,
clearly, explicitly, that a variable (say the peri-
meter of a polygon inscribed in a circle) can never
reach its limit (the circumference). The gain from
the standpoint of debating is very great ; a regular
inscribed polygon whose sides are steadily doubling
at set intervals of time, say, every half second,
presents a picture to the imagination which is
comparatively simple. The hopeless attempt of
imagining the limit as reached need not be made.
But this great gain is made at the expense of
generality. Robins descends to a very special type
of variation which is not the variation encountered
in ordinary mechanics ; it is an exceedingly artificial
variation. According to Robins's definition, Achilles
never caught the tortoise. It would not be difficult
to assume a time rate in the doubling of the sides
of a polygon inscribed in a circle, so that the cir-
cumference is reached. Thus, let the first doubling
of the number of sides take place in I second, the
second doubling in \ a second, the third in J a
second, and so on. It is easy to see that under
this mode of variation the polygons do reach the

98 LIMITS AND FLUXIONS

limit, the circumference. The process here tran-
scends our power of imagination, but lies within the
limits of reason. We are dwelling upon this point
because of its extreme importance in the history of
the theory of limits.

1 1 8. Robins constructs upon his first definition
the theorem, "that, when varying magnitudes keep
constantly the same proportion to each other, their
ultimate magnitudes are in the same proportion."
As a corollary of this he states "that the ultimate
magnitudes of the same or equal varying magni-
tudes are equal."

II. Definition: "If there be two quantities,
that are (one or both) continually varying, either
by being continually augmented, or continually
diminished ; though the proportion, they bear to
each other, should by this means perpetually vary,
but in such a manner, that it constantly approaches
nearer and nearer to some determined proportion,
and can also be brought at length in its approach
nearer to this determined proportion than to any
other, that can be assigned, but can never pass it :
this determined proportion is then called the ulti-
mate proportion, or the ultimate ratio of those
varying quantities."

Theorem: "To this definition of the sense, in
which the term ultimate ratio, or ultimate pro-
portion is to be understood, we must subjoin the
following proposition : That all the ultimate ratios
of the same varying ratio are the same with each
other,"

JURIN v. ROBINS AND PEMBERTON 99

119. Robins remarks thereupon that attempts at
the exposition of this method, so far as it depends
upon his first definition, were made by Lucas
Valerius in a treatise on the centre of gravity, and
by Andrew Tacquet in a treatise on the cylindrical
and annular solids ; but the development involving
his second definition was first made by Newton.
There are a number of writers, not mentioned by
Robins, who might be cited as forerunners in the
theory of limits ; such, for instance, as Gregory
St. Vincent and Stevin.

Newton's definition of momenta as the momentane-
ous increments or decrements of varying quantities,
" may possibly be thought obscure." Robins eluci-
dates thus: "In determining the ultimate ratios
between the contemporaneous differences of quanti-
ties, it is often previously required to consider each
of these differences apart, in order to discover, how
\ much of those differences is necessary for expressing
I that ultimate ratio" (§ 154). For instance, A^ + B#
only, and not the whole increment h.b-\-¥>a + ab, is
called the momentum of the rectangle under A,B.

1 20. Of this Discourse, a long account of twenty-
six pages, written by Robins himself, although his
name does not appear,1 was given in The Present
State of the Republick of Letters, London, October,
1735, in which it is staged that Robins wrote his
Discourse with the view of removing the doubts
which had lately arisen concerning fluxions and

1 This account is republished in the Mathematical Tracts of the late
Benjamin Robins, edited by James Wilscn, London, 1761, vol. ii, p. 78.

ioo LIMITS AND FLUXIONS

prime and ultimate ratios ; that Robins carefully
distinguished both these methods from the method
of indivisibles and also from each other. After
an historical excursion viewing the works of the
ancients, of Cavalieri and Wallis, the introduction
by Newton of the concept of motion is taken up.
"If the proportion between the celerity of increase
of two magnitudes produced together is in all parts
known," then <c the relation between the magnitudes
themselves must from thence be discoverable." This
is the basis for fluxions. The "method of prime
and ultimate ratios proceeds entirely upon the
consideration of the increments produced." - By it
Newton avoids "the length of the ancient demon-
strations by exhaustions," on which, according to
Robins, the method of fluctions rests. "Newton
did. not mean, that any point of time was assign-
able, wherein these varying magnitudes would
become actually equal, or the ratios really the
same ; but only that no difference whatever could
be named, which they should not pass." Newton's
term momentum is used simply for greater brevity,
hence need not be considered. Newton's descrip-
tion is capable of an interpretation too much
resembling the language of indivisibles — in fact, he
sometimes did use indivisibles at first ; Robins has
freed the doctrine from this imputation in a manner
that "shall agree to t.ne general sense of his
[Newton's] description."

JURIN v. ROBINS AND PEMBERTON 101

Jurirfs Review of his own Letters to Berkeley

121. In the November, 1735, number of the
Republick of Letters, Philalethes Cantabrigiensis
(Jurin) appears with an article, Considerations upon
some passages contained in two Letters to the Author
of the Analyst. The two letters in question are the
two replies Jurin himself had made to Berkeley.
The article is really a reply to Robins, though
Robins's name is not mentioned. Jurin claims to
have adhered strictly to Newton's language ; some
other defenders of Newton, says he, are guilty of
departing from it. Their objections to his own
defence are threefold :

" I. My explication of Lemma i, Lib. I, Princip."
See our §§4, 6, 8.

"II. The sense of the Scholium ad Sect. I,
Libr. I, Princ., particularly as to,

" i. The doctrine of Limits, 2. The meaning
of the term evanescent, or vanishing."
See our §§ 10-15.

4 'III. The sense of Lemma 2, Lib. II, Princip."
See our §§ 16-19.

122. As to the first objection, Jurin insists that
Newton's words fiunt ultimo aquales mean that the
quantities "do at last become actually, perfectly,
and absolutely equal." He takes the hands of a
clock between n and 12. The arcs traced by the
hands " i. Constantly tend to equality, 2. During
an hour, 3. And will come nearer to one another
than to have any given difference, 4. Before the

102 LIMITS AND FLUXIONS

end of the hour ; ... at the end of the hour, the
two quantities must become equal." Further, "by
taking the consideration of a finite time, Sir Isaac
Newton is 'able to assign a point of time, at which
he can demonstrate the quantities to be actually
equal." Consider, says Jurin, the ordinate to a
point of a hyperbola and that ordinate continued
to the asymptote : they do not become equal in
a finite time; Newton's Lemma is, "with great
judgment, so worded on purpose, as necessarily to
exclude this and such like cases." Thus Newton's
inscribed and circumscribed rectangles of Principia,
Lib. I, Sec. i, Lemma 2 (fig. i in our § 9), were
thought by Nieuwentiit and others never to be
capable of coincidence with the curve (say the
quadrant of a circle) ; but Jurin assumes the varia-
tion to be of such a nature that the limit
is actually reached, as demanded by Newton's
Lemma. For, suppose a point to move on the
horizontal radius from the circumference to the
centre A in one hour ; suppose also that, when
this moving point is at B on that radius, there
be two rectangles described upon AB (one in-
scribed, the other circumscribed), and that upon
every other part of the horizontal radius that is
equal to AB, namely the parts BC, CD, DE, taken
in order, rectangles be similarly erected "at the
same point of time," then as the moving point
nears the centre, the rectangles diminish in size
and increase in number, and they must together
become equal to the quadrant at the end of the

JURIN v. ROBINS AND PEMBERTON 103

hour. Jurin points out that he has introduced
here all the suppositions of Newton's first Lemma,
namely that, (i) the two figures tend constantly to
equality, (2) during one hour, i.e. a finite time, (3)
and come nearer to one another than to any given
difference, (4) before the end of the hour, i.e.
before the end of a finite time. Jurin continues :

"If any man shall say, that a right-angled
figure, inscribed in a curvilineal one, can never be
equal to that curvilineal figure ; much less to
another right-lined figure, circumscribed about the
curve ; I agree with him. I am ready to own that,
during the hour, these figures are one inscribed,
and the other circumscribed ; that neither of them
is equal to the curvilineal figure, much less one to
another. But then, on the other hand, it must be
granted me, that, at the instant the hour expires,
there is no longer any inscribed or circumscribed
figure ; but each of them coincides with the curvi-
lineal figure, which is the limit, the limes curvi-
lineus, at which they then arrive."

123. Jurin thereupon proceeds to Lemma 7 of
Book I, Section I in Newton's Principia, which, he
says, requires additional consideration. It relates
to fig. 4, where ACB is any arc and "the points A
and B approach one another and meet." Lemma 7,
in Andrew Motte's translation, reads as follows : —

4 'The same things being supposed; I say, that,
the ultimate ratio of the arc, chord, and tangent,
any one to any other, is the ratio of equality."

Jurin says that here the chord AB, the arch ACB,

104

LIMITS AND FLUXIONS

FIG. 4.

and the tangent AD come to vanish when B reaches
A, and their last ratio is unity. Newton "directs
our imagination, not to these vanishing quantities
themselves, but to others which are proportional
to them, and always preserve a
finite magnitude," such as A£,
the arch A^, Kd. Since at the
instant when A and B coincide,
"the angle BAD, or b&d, will
vanish ; it is easy to conceive
that, . . . the chord Ab must
coincide with the tangent A.d,
. . . consequently, AB, AD
must likewise, at the same instant of time, arrive
at the same proportion of a perfect equality. "

124. Proceeding to the last Scholium in Book I,
Section I of the Principia, Jurin starts by defining
the word limit. "I apprehend therefore that, by
the limit of a variable quantity, is meant some
determinate quantity, to which the variable quantity
is supposed continually to approach, and to come
nearer to it than to have any given difference, but
never to go beyond it. And by the limit of a
variable ratio, is meant some determinate ratio, to
which the variable ratio is supposed continually to
approach, and to come nearer to it than to have any
given difference, but never to go beyond it. By
arriving at a limit I understand Sir Isaac Newton
to mean, that the variable quantity, or ratio,
becomes absolutely equal to the determinate quan-
tity, or ratio, to which it is supposed to tend."

fURIN v. ROBINS AND PEMBERTON 105

With unusual lucidity, for that period, Jurin says
on the subject of limits : ' ' Now whether a quantity,
or ratio, shall arrive at its limit, or shall not arrive
at it, depends entirely upon the supposition we
make of the time, during which the quantity, or
ratio, is conceived constantly to tend or approach
towards its limit." If we assume the approach to
be made in a finite time, the limit is reached, other-
wise it is not reached. Of a variable which ' ' can
never attingere limitem " Newton gives one illustra-
tion at the end of the Scholium : that of two
quantities having at first a common difference and
increasing together by equal additions, ad infinitum.
Since they can never be really and in fact increased
ad infinitum, says Jurin, their ratio cannot arrive at
its limit. What Newton wanted to meet was the
objection, "that if the last ratio's of evanescent
quantities could be assigned, the last magnitudes
of those quantities might likewise be assigned."
Newton says No, * ' for those last ratio's, with which
the quantities vanish, strictly speaking, are not the
ratio's of the last quantities . . . but they are the
limits" which those ratios can never "arrive at,"
"before the quantities are diminished ad infinitum."
As to the sense in which Newton uses the word
evanescent or vanishing, in the Scholium under
consideration, Jurin inclines to the view that "both
imply one single instant, or point of time."

125. In the Principia, Book II, Section 2,
Lemma 2 (our §§ 16-19), Newton defines moment
as " a momentaneous increment, or decrement, of a

106 LIMITS AND FLUXIONS

flowing quantity, proportional to the velocity of the
flowing quantities." According to Jurin, Newton
puts a, b, c to signify either the moments, or the
velocities, of the flowing quantities A, B, C.
Leibniz looks upon them as differences. Newton,
says Jurin, never used indivisibles, and his method
to find the differences of variable quantities is not
"rigorously geometrical," but is more rigorous
than the treatment given by Leibniz.

Robins's Rejoinder

1 26. Robins replied in the Republick of Letters
for December, 1735, in a Review of some of the
Principal Objections that have been made to the
Doctrine of Fluxions and Ultimate Proportions ; with
some Remarks on the different Methods that have been
taken to obviate them. Robins does not here men-
tion Philalethes any more than the latter referred
directly to Robins. The objections to fluxions,
says Robins, are levelled at Newton's expression,
fluxiones sunt in ultima ratione decrementorum evan-
escentium vel prima nascentium. ' ' Which being
usually thus translated, that fluxions are in the
ultimate ratio of the evanescent decrements, or in
the first ratio of the nascent augments, it has from
hence been ask'd, what these nascent or evanescent
augments are ? " There are difficulties of interpre-
tation, whether the augments have quantity or
have not. One way out of this difficulty which
has been pointed out, is to say : "the limit of the
proportion that the decrements bear to each other

JURIN v, ROBINS AND PEMBERTON 107

as they diminish, is the true proportion of the
fluxions" (p. 438). Here a new difficulty arises:
Does the varying ratio reach its limit " actually,
perfectly, and absolutely," or does it not? On the
understanding that it does not reach the limit, "all
that has at any time been demonstrated by the
ancient method of exhaustions may be most easily
and elegantly deduceoV' Rigour of demonstration
does not require ultimate coincidence. ''Coinci-
dence of the variable quantity and its limit, could
it be always prov'd, would yet bring no addition to
the accuracy of these demonstrations" (p. 441).
Hence, "why to the natural difficulty of these
subjects should the obscurity of so strained a con-
ception be added ? " Is this view a correct inter-
pretation of Newton ? A literal translation of his
Lemma I, Section I, Book I, Principia (see our
§§ 4, 6, 8), is: "Quantities, and the ratios of
quantities, that during any finite time constantly
approach each other, and before the end of that
time approach nearer than any given difference,
are ultimately equal." What is the meaning of
"given difference"? If it be a ''difference first
assigned" according to which the degree of approach
of these quantities may be afterwards regulated ;
then . . . ratios, and their limits, tho' they do
never actually coincide, will come within the de-
scription of this Lemma ; since the difference being
once assign'd, the approach of these quantities may
be so accelerated, that in less than any given time
the variable quantity, and its limit, shall differ by

io8 LIMITS AND FLUXIONS

less than the assign'd difference." Here Robins
expresses the idea that the rapidity of approach
toward the limit can be arbitrarily altered, yet he
does not apparently perceive — certainly he does not
admit — that this rapidity may be altered in such a
way that the variable actually does reach its limit.
On the contrary, he maintains that " where the
approach is determin'd by a subdivision into parts,"
"it is obvious, that no coincidence can ever be
obtain'd." A coincidence such as Philalethes
explains in the case of rectangles circumscribed
and inscribed in a curve, if it could take place, is
not a coincidence such as Newton intended, for
Newton did not in this instance use motion, but
continual subdivision. Robins tries to establish
his view of the matter by giving an instance of
erroneous results being deduced by letting the
variable reach its limit. He takes a hyperbola
and revolves its principal axis in the plane of the
curve, around the point of intersection of this axis
and an asymptote, until the two lines coincide.
At the end "the hyperbola coincides with the
asymptote," which is "absurd." As a matter of
fact there is no absurdity. In fix*1 — a2y2 = a2&2,
the slope of the asymptote is m = b / a. Robins's
process amounts to making m = o, which gives a
real locus when b = <?, namely the locus j2 = o. The
only objection lies in still calling the final curve a
"hyperbola."

JURIN v. ROBINS AND PEMBERTON 109

The Debate Continued

1 27. Robins's article was followed in the January,
1736, number of the Republick of Letters by
Philalethes's Considerations occasioned by a Paper in
the last Republick of Letters, concerning some late
Objections against the Doctrine of Fluxions ', and the
different Methods that have been taken to obviate
them. Jurin denies having said that there was
an "intermediate state" between augments being
"any real quantity" and being "actually vanished";
he says he gave Newton's declaration that " their
magnitude cannot be assigned or determined. " Such
intermediate magnitudes, in Jurin's opinion, cannot
be "represented to the mind," but their ratio can
be represented to the mind, by contemplating the
ratio, " not in the vanishing quantities themselves,
but in other quantities permanent and stable,
which are always proportional to them " (p. 76).
As to Newton's Lemma i in Section I, Book I of
the Principia, if the great author meant to conclude,

(that the quantities "approach nearer than any
given difference," then he first supposed what he
would prove, and proved only what he had before
supposed. Of this he could not be guilty. Besides,
Newton's words,1 "fiunt aequales," do absolutely
subvert such an interpretation. Jurin says that
he does not claim that coincidence is necessary for

» rigorous proof; he admits that in Robins's treat-

ment of prime and ultimate ratios, coincidence is

1 Newton's words are " fiunt ultimo aequales." See our § 4.

no LIMITS AND FLUXIONS

not necessary ; only, Robins's method is not that of
Newton. To establish this last point, Philalethes
quotes Newton's lemma in Latin, then prints Robins's
and his own translation of it. In case of variation,
the upper line is Robins's translation, the lower is
Jurin's : —

Quantities, and \ ' \ ratio's of quantities, that
^ also J

. ( approach each other ; \
during any finite time constantly \ .

{ tend to equality, }

and before the end of that time approach nearer
( than any given difference, are ultimately equal. \

\ to one another than to have any given difference, do \
at last become equal. J

It is not clear to Jurin what Robins means by
"are ultimately equal," nor can Jurin conceive
"how quantities, which do never become actually
equal, . . . can come within the description of a
Lemma, which Lemma expressly affirms, that they
become equal. " Fiunt ultimo cequales means ' f become
at last equal." Jurin quotes different places in the
Principia which support his point. He denies that
Newton proceeds, in the case of inscribed and
circumscribed rectangles, by continual divisions of
the base of the figure, and gives references in
support of his contention. Of interest are the
following admissions made by Jurin (p. 87): "This
equality therefore we are obliged to acknowledge,
although we should not be able, by stretch of
imagination, to pursue these figures, and, as it
were, to keep them in sight all the way, till the

JURIN v. ROBINS AND PEMBERTON in

very point of time that they arrive at this equality.
For a demonstrated truth must be owned, though
we do not perfectly see every step by which the
thing is brought about."

"We have therefore no occasion for the delinea-
tion of a line less than any line that can be assigned.
We acknowledge such delineation to be utterly
impossible ; as likewise the conception of such a
line, as an actually existing, fixed, invariable,
determinate quantity." Jurin here begins to dis-
avow infinitesimals. "I am very free to own that
Sir Isaac Newton does not always consider this
coincidence, or rather equality, of the variable
quantity, or ratio and its ultimate, as necessary in
his method."

128. The debate between Jurin and Robins had
reduced itself by this time, not so much to the
discussion of the logical foundations of fluxions, as
to the discussion of what Newton's own views on
the subject had been. Robins prepared a long
paper on the subject for the April, 1736, issue of
the Republick of Letters, under the title : A Dis-
sertation shewing, that the Account of the doctrine of
Fluxions, and of prime and ultimate ratios, delivered
in a treatise entitled, ' A discourse concerning the
nature and certainty of Sir Isaac Newton's methods
of fluxions, and of prime and ultimate ratios, ' is
agreeable to the real sense and meaning of their great
inventor. The paper covers 45 pages. Robins
repeats the fundamental definitions and historical
statements given in his earlier papers, and directs

ii2 LIMITS AND FLUXIONS

some attacks against Berkeley. To set forth the
views of Newton, quotations are made from his
works. He quotes from the Introduction to the
Quadratura Curvarum (see our §§ 27-42). From
the Quadratura Curvarum itself he quotes:

" Quantitates indeterminatas, ut motu perpetuo
crescentes vel decrescentes, id est, ut fluentes vel
defluentes, in sequentibus considero, designoque
literis z, y, x, v, et earum fluxiones, seu celeritates
crescendi noto iisdem literis punctatis. Sunt et
harum fluxionum fluxiones, sive mutationes magis
aut minus celeres, quas ipsarum 2, y, x, v fluxiones
secundas nominare licet," etc.

Robins quotes also from the anonymous account
of John Collins's Commercium Epistolicum, which
figures so prominently in the controversy between
the followers of Newton and of Leibniz. This
account was published in the Philosophical Trans-
actions, vol. xxix, for the years 1714, 1715, 1716,
of the Royal Society of London, of which Robins
was a member. Robins goes on the assumption
that the anonymous article was written by Newton
himself, an assumption denied by no one at that
time or since, though Jurin in a reply wants to
know on what authority Newton's authorship is
asserted. Robins quotes as follows (see our § 47):

"When he [Newton] considers lines as fluents
described by points, whose velocities increase or
decrease, the velocities are the first fluxions, and
their increase the second. "

129. Robins says that Berkeley, "for the support

JURIN v. ROBINS AND PEMBERTON 113

of his objections against this doctrine [of fluxions],
found it necessary to represent the idea of fluxions
as inseparably connected with the doctrine of
prime and ultimate ratios, intermixing this plain
and simple description of fluxions with the terms
used in that other doctrine, to which the idea of
fluxions has no relation : and at the same time
by confounding this latter doctrine with the
method of Leibniz and the foreigners, has proved
himself totally unskill'd in both. These two
methods of Sir Isaac Newton are so absolutely
distinct, that their author had formed his idea of
fluxions before his other method was invented, and
that method is no otherwise made use of in the
doctrine of fluxions, than for demonstrating the
proportion between different fluxions. For, in Sir
Isaac Newton's words [see our §§ 29, 36], as the
fluxions of quantities are nearly proportional to
the contemporaneous increments generated in very
small portions of time, so they are exactly in the
first ratio of the augmenta nascentia of their fluents.
With regard to this passage the writer of the
Analyst has made a two-fold mistake. First, he
charges Sir Isaac Newton, as saying these fluxions
are very nearly as the increments of the flowing
quantity generated in the least equal particles of
time. Again, he always represents these augmenta
nascentia, not as finite indeterminate quantities, the
nearest limit of whose continually varying pro-
portions are here called their first ratio, but as
quantities just starting out from non-existence, and

ii4 LIMITS AND FLUXIONS

yet not arrived at any magnitude, like the infini-
tesimals of differential calculus. But this is con-
trary to the express words of Sir Isaac Newton,
who after he had shewn how to assign by his
method of prime and ultimate ratios the proportion,
that different fluxions have to one another, he thus
concludes. In finitis autem quantitatibus Analysin
sic instituere et finitarum nascentium vel evanescentium
rationes primas vel ultimas investigare consonum est
geometries vetej-um: et volui ostendere, quod in methodo
fluxionum non opus sit figuras infinite pat'vas in
geometriam introducere." (See our §§ 33, 41.)

130. Robins proceeds to explain that the method
of prime and ultimate ratios is "no other than the
abbreviation and improvement of the form of
demonstrating used by the ancients on the like
occasions." It has nothing to do with infinitely
small quantities, which have led into error not only
Leibniz in studying the resistance of fluids and the
motion of heavenly bodies, but also Bernoulli like-
wise in the resistance of fluids and in the study of
isoperimetrical curves. Such infinitely small quanti-
ties led Parent to make wrong deductions. It was
argued that because a heavy body descends through
the chord of a circle terminating at its lowest point
in the same time as along a vertical diameter, " the
time of the fall through the smallest arches must be
equal to the time of the fall through the diameter."
To relieve Newton of the suspicion of not being
free from the obscure methods of indivisibles,
Robins says he [Robins] defined an "ultimate

JURIN v. ROBINS AND PEMBERTON 115

magnitude " and "ultimate ratio" as limits. This
exposition Robins had given in full in his Discourse.
The difference of interpretation of Newton's Lemma
I in the Principia (Book I, Section i), given by him-
self and by Jurin, arises from Jurin's misinterpre-
tation of Newton's word given. He "supposes it
to stand for assignable; whereas it properly signi-
fies only what is actually assigned." Jurin claims
that by our interpretation, "Newton is rendred
obnoxious to the charge of first supposing what
he would prove " (p. 307). Robins says in reply
that the statement, quantities which "are perpetu-
ally approaching each other in such a manner, that
any difference how minute soever being given, a
finite time may be assigned, before the end of
which the difference of those quantities or ratios
shall become less than that given difference," is an
obvious but not an identical proposition. Robins
argues, "that Sir Isaac Newton had neither
demonstrated the actual equality of all quantities
capable of being brought under this lemma, nor
that he intended so to do " (p. 309) ; when quanti-
ties ' ' are incapable of such equality, the phrase of
ultimately equal must of necessity be interpreted in
a somewhat laxer sense," as in Principia, Book I,
Prop. 71, "pro aequalibus habeantur, are to be
esteemed equal." When Newton says that the
number of inscribed parallelograms should be
augmented in infinitum, he does not mean that
the number becomes infinitely great, but that they
are augmented endlessly. The nature of the motion

n6 LIMITS AND FLUXIONS

assumed by Jurin to explain how the limit may be
reached is excessively complex. Moreover, ' ' to
assert that any collection of these inscribed and
circumscribed parallelograms can ever become
actually equal to the curve, is certainly an impro-
priety of speech, . . . the essence of indivisibles
consists in endeavouring to represent to the mind
such inscribed or circumscribed figure, as actually
subsisting, equal to the curve" (p. 312). Our
interpretation "thus removes this doctrine quite
beyond the reach of every objection" (p. 315).
Robins argues that Newton's ultima rationes,
quibuscum quantitates evanescunt are not rationes
quantitatum ultimarum ; but only limits, to which
the ratios of these quantities, which themselves
decrease without limit, continually approach ; and
to which these ratios can come within any differ-
ence, that may be given, but never pass, nor even
reach those limits" (p. 316). f< Newton has
expressly told us, that the quantities, he calls
nascentes and evanescentes, are by him always con-
sidered as finite quantities" (p. 321).

131. The momenta of quantities occur in Newton's
De analyst per czquationes numero terminorum infinitas,
drawn up in 1666. Newton says "that he there
called the moment of a line a point in the sense of
Cavalerius, and the moment of an area a line in the
same sense " (see our § 47), that " from the moments
of time he gave the name of moments to the momen-
taneous increases, or infinitely small parts of the
abscissa and area generated in moments of time . . .

JURIN v. ROBINS AND PEMBERTON 117

because we have no ideas of infinitely little quantities,
he introduced fluxions into his method, that it might
proceed by finite quantities as much as possible."
Prime and ultimate ratios he introduced later.
Newton says in that place that in his proofs he
uses o for a finite moment of time, though some-
times, for brevity, he supposes o infinitely little.
Thus Newton used o in two senses ; in the fluxions
published in 1693 in Wallis's algebra, o is used in
the sense of indivisibles ; in 1704 he gave it a
second signification in the Quadratura Curvarum.
Robins sums up his dissertation thus: " Hence it
is very manifest, that as Sir Isaac Newton used at
first indivisibles, so he soon corrected those faulty
notions by his doctrine of fluxions, and afterwards
by that of prime and ultimate ratios. And all the
absurdity of expression, and all the inconsistency
with himself charged on him by the author of the
Analyst, arises wholly from mis-representation."
This paper was badly arranged and below the level
of Robins's earlier contributions.

132. Robins's long paper in the Republick of
Letters was followed in the July and August
(vol. xxviii, 1736) numbers by Considerations
upon some passages of a Dissertation concerning the
Doctrine of Fluxions, published by Mr. Robins in the
Republick of Letters for April last, by Philalethes
Cantabrigiensis. The paper extends over 136 pages,
and could not be easily accommodated in a single
number. From now on the disputants, particularly
Jurin, are no longer in a calm frame of mind. The

n8 LIMITS AND FLUXIONS

debate is one over words and ceases to be illumi-
nating. Their judgments were perverted by the
heat of controversy. Even theological or political
controversies could not easily surpass the verbosity
and haze exhibited here.

Jurin's first objection to Robins's last analysis is
the statement that the method of fluxions has no
relation to the method of first and last ratios ; Jurin
quotes from Newton in support of his contention.
The charge that he (Jurin) represents augmentia
nascentia not as finite, but as just starting out of
non-existence, "like infinitesimals of the differential
calculus," Jurin denies, saying : Leibniz's differ-
entials "are fixed, determinate, invariable"; he
himself has represented the nascent augments as
"quantities just starting out from non-existence,
and yet not arrived at any magnitude, and not as
finite quantities" (p. 52), and quotes Newton in
support of this view. According to the article in
the Philosophical Transactions, No. 342, attributed
to Newton, moments are represented "by the
rectangles under the fluxions and the moment o " ;
"in his calculus there is but one infinitely little
quantity represented by a symbol, the symbol o : it
is also said, Prick'd letters never signify moments,
unless when they are multiplied by the moment o
either exprest or understood to make them infinitely
little, and then the rectangles are put for moments."
Jurin charges that Robins has now published four
different interpretations of Newton's much-discussed
lemma. Newton's phrase, fiunt ultimo cequales, the

JURIN v. ROBINS AND PEMBERTON 119

use of the words <( perpetually " and "endlessly,"
"the last difference," are again discussed at length.
Jurin quotes from Robins a passage which appears
to show that * ' Mr. Robins is now of opinion, that
Sir Isaac's demonstration is applicable to such
quantities, as at last become actually equal, as well
as to quantities, which only approach without limit
to the ratio of equality " (p. 67) ; therefore, the
lemma, "by Mr. Robins's own confession, maybe
taken in the sense I have always understood it in "
(p. 68). However, this is in direct conflict with
Robins's earlier assertions. In the discussion about
the inscribed rectangles, both Robins and Jurin
agree that if the ' ' base of the curve " (our abscissa)
be continually subdivided as in Euclid I 10 or V
10, it is manifest "that such subdivision can never
be actually finished " (p. 78) ; but Newton proceeds
differently — he supposes a line to be described by a
moving point. Jurin thereupon repeats exactly the
argument in Zeno's "Dichotomy," though he does
not mention Zeno, to show that a point moving
across the page in, say, one hour passes over 1/2
of the distance, then over I / 4 of it, then over
1/8, i / 1 6, etc., and insists that "all the possible
subdivisions of the line " will be " actually finished "
and "brought to a period at the end of the
hour." This is given in support of his previous
argument that the rectangles inscribed in a curve
may reach the limit. " If Mr. Robins will tell me,
that the imagination cannot pursue these parallelo-
grams to the very end of the hour, I may ask him,

120

LIMITS AND FLUXIONS

whether the imagination can any better pursue the
subdivision of the line, or even of the hour itself, to
the end of the hour, which subdivisions he must
own to be brought to a period by the end of the hour.
But there is no need to strain our imagination, to
labour in every case, or indeed in any case, after
some idea of motion however intricate ; no subtle
inquiry is at all necessary, since we are obliged to
own the conclusion to be true and certain. ..."

" However, since Mr.
Robins is pleased to talk so
much about straining our
imagination, ... let us see,
if we cannot find some plain
and easy way of represent-
ing to the imagination, that
actual equality, at which the
inscribed and circumscribed
figures will arrive with each
other, and with the curvi-
lineal figure, at the expiration of the finite time"
(p. in).

Let the curvilineal figure ABE equal in area the
rectangle with sides EA and AF. When the moving
point describing the base EA in a finite time is at C,
let the rectangle with the base EA and height Cd
be equal to the sum of the parallelograms inscribed
in ABE (not drawn in the figure) which stand on
CA and upon as many other adjoining parts of EA
as can be taken equal to CA. Let }Ldd be the curve
traced by the moving point d.

JURIN v. ROBINS AND PEMBERTON 121

Let the area of the rectangle with EA as
base and CD as height = sum of the circumscribed
parallelograms (not drawn in the figure) standing on
CA and upon as many other parts of EA as can be
taken equal to CA and adjoining to it ; also let
GDD be the curve traced by the movable point D.
Then as the curvilineal, the inscribed, and the
circumscribed figures are respectively equal to EA x
AF, EA x Cd, EA x CD, these figures must be pro-
portional to AF, Cd, and CD. These three lines
will * ' be equal to one another at the end of the
finite time." Now since Cd and CD approach each
other, during a finite time, within less than any
given distance before the end of that time, these
three lines will, by that Lemma, be equal to one
another at the end of the finite time. The limit is
reached (p. 1 14).

133. As a further illustration, Jurin takes a
rectilinear figure, the right triangle ABE, where
EA = AB = #, AF = Jtf, EC=^r, the point C mov-
ing from E to A as before. Upon AC as a base,
imagine an inscribed rectangle (height CH), and
a circumscribed rectangle (height CK). As in
the previous figure, imagine other inscribed and
circumscribed rectangles, standing upon as many
other parts of EA as can be taken equal to CA,
and adjoining to it in order. When CA is an
aliquot part of AE, then a x Cd is the sum of the
inscribed rectangles and a x CD is the sum of the
circumscribed rectangles, where Cd=x / 2, and
= a-x I 2. Let K</=CD. The ordinate Kd,

122

LIMITS AND FLUXIONS

drawn to the base BG, will be terminate by EF.
When CA is not an aliquot part of AE, if we
divide the base into as many parts as may be,
there will be left a portion E£, which, let us
call r. Then Cd=x+rx a — r-±(2a) and all these
ordinates will be bounded by
In the same way,
— x+ r^ a — r -^ (za),
and the ordinate will be
bounded by E^F. When
x=a, r vanishes, Cd= \ a and
Kd= \a. Hence the inscribed
and circumscribed figures do
then become equal to each
other, and to the triangle
ABE ; again, the limit is reached.

Jurin takes Robins to task for asserting that
" equality can properly subsist only between figures
distinct from each other." To Robins's query,
"Does Philalethes here suppose the truth of Sir
Isaac Newton's demonstrations to depend on this
actual equality of the variable quantity and its
limit?" Jurin answers, " I do . . . In the manner
Mr. Robins defines, and treats of prime and ultimate
ratios, I allow his demonstrations to be just without
this actual equality. But Sir Isaac Newton does
not define and treat of prime and ultimate ratios,
in the same manner with Mr. Robins ; nor are
Mr. Robins's demonstrations at all like Sir Isaac
Newton's demonstration" (p. 128). The inability
of our imagination to pursue the rectangles in reach-

JURIN v. ROBINS AND PEMBERTON 123

ing the limit is no valid argument against the con-
tention that the limit is reached ; even in the
ancient geometry there are demonstrated truths
that lie beyond the reach of the imagination, as
for instance, that three cones may equal a cylinder,
all of the same base and height (p. 130). The
meaning of moment^ a truly difficult concept, is dis-
cussed again, Jurin holding that Newton took it as
"a mcmentaneous increment, . . . less than any
finite quantity whatsoever, and proportional to the
velocity of the flowing quantity," while Robins seem-
ingly claimed that Newton meant them to be finite
quantities (p. 151). With respect to Newton's
early use of the infinitely little, Jurin and Robins
were in disagreement, and Robins was in our
opinion nearer the truth. Robins claimed that
Newton at first used infinitely little quantities ; that
afterwards he improved his method by discarding
them ; Jurin claimed that Newton's alleged absurdity
of expression and inconsistency with himself, as
charged by Berkeley and others, "arises wholly
from misinterpretation, or misunderstanding him "

(p. 179).

134. Jurin's article appeared in the July and
August numbers, 1736, of the Republick of Letters.
Robins could not wait in patience until the entire
article of Jurin had been printed. In the August
number he replies to the part of Jurin's article that
had appeared in the July number. The August
number was given up to Jurin and Robins, to the
entire exclusion of all other articles and of the usual

i24 LIMITS AND FLUXIONS

book reviews. On the last page of the August
number, the editor apologises to the readers and
assures them ' * they shall hereafter have no occa-
sion to complain upon this head." In Robins's
reply,1 both " Robins" and " Philalethes " appear
in the third person, as if the writer were some out-
sider. Robins says : ' * Newton does not intermix
his simple and plain description of fluxions with the
terms used in the doctrine of prime and ultimate
ratios ; for his description of fluxions is contained
in the two first paragraphs of his Introduction to
the Quadratures, in which no terms of the other
doctrine occur " (p. 89). The Lemma is, of course,
taken up again, Robins claiming his interpretation
legitimate, "for two quantities may constantly tend
to equality during some finite space of time, and
before the end of that time come nearer together
than to have any difference, which shall be given ;
and yet at the end of that time have still a real
difference," while Jurin's interpretation was not
"any difference that shall be given," but "any
assignable difference," which would mean that the
limit must be reached. Mr. Robins says (p. 97) :
"It is not difficult to assign a very probable reason,
which led Sir Isaac Newton to the use of this
expression [fiunt ultimo sequales], for before him it
had not been unusual for geometers to speak of the
last sums of infinite progressions, which is an ex-

1 "Remarks on the Considerations relating to Fluxions, etc., that
were published by Philalethes Cantabrigiensis in the Republick of
Letters for the last month," Republick of Letters, August, 1736,
pp. 87-110.

JURIN v. ROBINS AND PEMBERTON 125

pression something similar to this. Surely here no
one will pretend, that an infinite number of terms
can in strict propriety of speech, and without a
figure, be said to be capable of being actually
summed up and added together." Robins makes
the only direct reference that was made in this
debate to Zeno's paradoxes. He mentions Achilles
and the Tortoise, but in a manner devoid of interest.
Referring to the line which Jurin supposes traced in
one hour, Robins says : " Perhaps it may be easiest
understood by comparing the present point with
the old argument against motion from Achilles and
the Tortoise. It is impossible to pursue in the
imagination their motion by the means proposed
in that argument to the point of their meeting,
because the motion of each is described by the
terms of an infinite progression." Robins does
not seem fully to realise that Achilles and the
Tortoise present a case in which a variable reaches
its limit.

135. The editor of the Republick of Letters
permitted the two disputants to continue their
wranglings in an Appendix to the September issue.1

Philalethes's attempt to represent to the imagina-
tion the actual equality at which the inscribed and
circumscribed figures will arrive with each other, and
with the curvilinear figure, is criticised by Robins

1 An Appendix to the Present State of the Reptiblick of Letters for
the Month of September, 1736. Being Remarks on the Remainder of
the Considerations relating to Fluxions , etc.t that was published by
Philalethes Cantabrigiensis in the Republick of Letters for the last
Month. To which is added by Dr. Pemberton a Postscript occasioned by
a passage in the said Considerations. London, 1736.

126 LIMITS AND FLUXIONS

on the ground that the continued curve "is not to
be described, but by an endless number of para-
bola's " (of which the curve is the envelope) ; thus,
Philalethes gave "as an equation expressing the
nature of a single curve, one which in reality
includes an infinite series." " Philalethes supposed
a last form of the inscribed figures, that was equal
to the curve." Robins observed "that equality
implies the things, which have that property, to be
distinct from each other. For to say a thing is
equal to itself is certainly no proper expression."
But "there is no such last form distinct from the
curve," as Philalethes admits ; hence Philalethes
"gives up the point."

136. In the Principia, Newton does not deliver
the doctrine of fluxions, but the doctrine of prime
and ultimate ratios. "The understanding of this
book is attended with difficulty." The expression
ultima summa is defective : " Can any sum of a set
of quantities, whose number is supposed infinite, in
strict propriety of speech be called their last sum ? "
Later, Robins says : " Let Philalethes reconcile
the actual arrival of these quantities to the ratio
supposed, and at the same instant vanishing away.
Is not this saying, that the two quantities become
nothing, and bear proportion at the same instant of
time ? " (p. (14)). Philalethes "has thought himself
unjustly accused by Mr. Robins of supposing a
nascent increment to be some intermediate state of
that increment between its finite magnitude, and
its being absolutely nothing. To have proved this

JURIN v. ROBINS AND PEMBERTON 127

assertion groundless he ought to have shown, that
this definition does not attempt at describing such
an intermediate state" (p. (15)). Robins asserts :
"Whoever has read Sir Isaac Newton's Lectiones
Optic<2y and will deny, that he has at any time
made use of indivisibles, must be very much a
stranger to that doctrine, and to the style of those
writers who follow it" (p. (19)). "What reflexion
is it upon Sir Isaac Newton to suppose, that he
made use of the methods he had learned from others
before he had invented better of his own : or that
in an analysis of a problem for dispatch he still
continued to make use of such methods, when he
conceived it would create no error in the conclusion ?
Has not Sir Isaac Newton said this of himself, and
has Mr. Robins said anything more?" (p. (15)).
"Does Philalethes here mean, that a quantity can
become less than any finite quantity whatsoever,
before it vanishes into nothing? If not, then the
point is given up to Mr. Robins, who only contends,
that vanishing quantities can never by their diminu-
tion be brought at last into any state or condition,
wherein to bear the proportion called their ultimate :
if otherwise, since Philalethes supposes . . . that
it is nonsense, that it implies a contradiction to
imagine a quantity actually existing fixed, deter-
minate, invariable, indivisible, less than any finite
quantity whatsoever ; because this imports as much
as the conception of a quantity less than any
quantity, that can be conceived : how can a quantity
supposed to be less than any finite quantity whatso-

128 LIMITS AND FLUXIONS

ever be rendered more the object of the conception
by being understood to be brought into this con-
dition by a constant diminution from a variable
divisible quantity?" (p. (20)). "Sir Isaac Newton
has introduced into use the term moment throughout
the whole second book of the Principia, and for no
other purpose than for the sake of brevity ; for his
doctrine of prime and ultimate ratios had been before
fully explained, and every proposition of the second
book might have been treated on without the use
of this term, though perhaps with a somewhat
greater compass of words " (p. (23)). " Mr. Robins
has endeavored to defend Sir Isaac Newton both
against the accusation of the author of the Analyst,
and the misrepresentation of Philalethes. He has
shown, that Sir Isaac Newton's doctrine of prime
and ultimate ratios has no connexion with indi-
visibles, and that, if he ever allowed himself in the
use of indivisibles, he knew that he did so, and did
not confound both the methods together, as the
author of the Analyst accuses him, and Philalethes
without knowing it has owned " (p. (27)). ' ' Had
Philalethes been versed in the ancients, and in the
later writers who have imitated them, he could
have been at no loss about the true sense of data
quavis differentia used by Sir Isaac Newton in his
first Lemma. For this expression is borrowed from
the writers, that made use of exhaustions " (p. (29)).
" What separates the doctrine of prime and ultimate
ratios from indivisibles is the declaration made in
the Scholium to the first Section of the Principia^

JURIN v. ROBINS AND PEMBERTON 129

that Sir Isaac Newton understood by the ultimate
sums and ratios of magnitudes no more than the
limits of varying magnitudes and ratio's ; and he
puts the defence of his method upon this, that the
determining any of these limits is the subject of
a problem truly geometrical. To insist, that the
variable magnitudes and ratio's do actually attain,
and exist under these limits, is the very essence of
indivisibles " (p. (34)).

Robins's reply in the August and September,
1736, numbers of the Republick of Letters is con-
densed in form, yet covers 61 pages. It is im-
possible for us to convey an adequate idea of
the amount of detail entering in the discussion.
Altogether Robins shows greater willingness to
admit that Newton's views were different at differ-
ent periods in his career, and that even Newton
may be guilty of modes of expression that are
not free from obscurity. Moreover, Robins speaks
in general with greater sincerity than his opponent.
But Jurin proves himself the superior of Robins
in adhering to a broader and more comprehensive
conception of variables and limits.

Pemberton enters the Debate

137. At this stage a new party enters the debate
— Henry Pemberton, who had studied medicine and
mathematics at Leyden and Paris, had been a friend
of Newton, and had edited the third edition of the
Principia. In an article following the one of Robins
in the " Appendix" (August and September 1736),

130 LIMITS AND FLUXIONS

Pemberton says : ' ' I . . . am fully satisfied, that
Mr. Robins has expressed Sir Isaac Newton's real
meaning." Pemberton quotes from Newton's Intro-
duction to the Quadrature of Curves about prime and
ultimate ratios (see our §§33, 42), and then remarks;
"Here Sir Isaac Newton expressly calls the quan-
titates nascentes and evanescentes, whose prime and
ultimate ratios he investigates, by the appellation of
finite. Now I desire Philalethes to reconcile this
passage with his notion of a ' nascent quantity
being a quantity not yet arrived at any assignable
magnitude how small soever.' And I must farther
ask Philalethes, whether he has not here attempted
to define a non-entity."

138. Robins's last article and Pemberton's rash
challenge led to another flow of words, covering
77 pages in the ''Appendix" to the Republick of
Letters for November, 1736, in an article by Jurin,
entitled Observations upon some Remarks relating to
the Method of Fluxions, published in the Republick of
Letters for August last, and in the Appendix to that
for September.

Jurin insists that "the method of fluxions, as it
is drawn up by Sir Isaac Newton, could not possibly
^formed before the method of first and last ratio's
was invented" (p. (6)).

Robins "takes no notice of the letter o being
used in the book of Quadratures, in the very same
sense as in the Analysis" (p. (8)). "That symbol
never denotes any quantity, but what, by a con-
tinual decrease, becomes infinitely little, i.e. less

JURIN v. ROBINS AND PEMBERTON 131

than any quantity, and at last vanishes into
nothing" (p. (8)).

"He is grossly mistaken in thinking, that
quantities, which, before the end of a finite time,
come nearer together than to have any assignable
difference, will therefore become equal before the
end of that time " (p. (12)). " I have clearly proved
in November and January last, that Sir Isaac
Newton designed no quantities or ratio's to be com-
prehended within the sense of this lemma, which
do not become actually equal " (p. (13)).

"Has then Mr. Robins, . . . offered to shew,
that any quantities or ratio's incapable of an actual
equality are compared in this lemma? I think not"
(p. (22)). In January, " I use the following words,
4 This determinate proportion of the finite quantities
a and e, is what I understand by the proportion of
the evanescent augments.' This, I say, ought to
have been attended to, before this charge against
me was renewed " (p. (24)). As regards the ratio
between the inscribed and circumscribed figures,
4 'have not I truly expressed it? If my expression
be too complex, let these great Geometers shew me
a simpler, if they can, and I will make use of that "
(p. (34)). Robins's argument about the last form of
parallelograms differing from the limiting curve is
defective in the minor of the syllogism: "Things
which are equal are distinct from each other." " Is
it," says Jurin, " the part of a candid and ingenious
adversary, to insist always upon the word equal, when
a more proper expression, as that of coinciding, has

132 LIMITS AND FLUXIONS

been used by his antagonist?" If his argument
is sound, "it will hold against my expression,
that the figures inscribed and circumscribed do at
last coincide with the curvilinear figure." Jurtn
claims "that if Mr. Robins's interpretation of the
first Lemma be admitted, Sir Isaac's demonstra-
tions, as they now stand, will not be accurate, nor
geometrically rigorous," for, "as they now stand,
the examples he has given in the several Lemmata
of the first Section, are of such quantities and
ratio's only, as do actually arrive at their respective
limits" (pp. (42) and (43)). "Mr. Robins and I
have been disputing some time, whether Sir Isaac
Newton used indivisibles. That Gentleman main-
tains that he used them ; and grounds his charge
upon the term infinitely little, which is sometimes to
be met with in Sir Isaac Newton's writings : but he
does not explain the meaning of that term, when
used either by Sir Isaac, or by the writers of indi-
visibles. I, on the contrary, distinctly explain
what I apprehend to be meant by it, both when
used by Sir Isaac Newton, and when used by the
writers of indivisibles. ... I supposed the writers
upon indivisibles, by an infinitely little quantity,
to mean a quantity actually existing, fixed, deter-
minate, invariable, indivisible, less than any finite
quantity whatsoever" (p. (73)). Robins quotes
Pascal and Barrow as using the term indefinite in
place of infinite, but the writers I quoted use infinite
and infinitely little. There is difference of usage
among followers of Cavalieri. "It is not denied,

JUR1N v. ROBINS AND PEM^ERTON 133

but that Sir Isaac Newton, by the term infinitely
little^ meant a quantity variable, divisible, that, by
a constant diminution, is conceived to become less
than any finite quantity whatsoever, and at last to
vanish into nothing. By which meaning all that is
faulty in the method of indivisibles, is entirely
avoided ; and that being allowed, the rest is only
a dispute about a word " (p. (74)).

Jurin declares in a "Postscript" that " to carry
on two controversies at once is more than I have
leisure for"; later "I intend to accept of Dr.
Pemberton's invitation " ; meanwhile Jurin inserts
an attestation of ''his learned friend Phileleutherus
Oxoniensis " to the effect that this friend is * ' fully
satisfied, Mr. Philalethes has expressed Sir Isaac
Newton's real meaning." The language of this
attestation follows exactly the language of Pember-
ton, except that Philalethes, and not Robins, is
now declared the correct interpreter of Newton.

139. In the December issue, 1736, of the Republick
of Letters ', Robins says in an "Advertisement" that
"since Philalethes has given loose to passion," he
"cannot think anything farther necessary for the
satisfaction of impartial readers " (p. 492), and now
takes "leave of Philalethes," but cannot resist a
few parting shots. Nor could Philalethes resist
making reply to this "Advertisement" in an
"Appendix" to the December number, 1736, of
the Republick of Letters, in which he expresses regret
' ' that so long a correspondence should end in dis-
content or ill humour." Jurin justifies the practice

134 LIMITS AND FLUXIONS

he exercised in this controversy of offering poetry
(usually in Latin) for the sake of readers who are
under necessity "of exercising their faith, rather
than their reason in this dispute," for "A verse
may catch him, who a sermon flies," and for the
sake of enlivening the subject for others, "who
are judges of the dispute."

140. In this December " Appendix " Jurin then
contributes A Reply to Dr. Pembertoris Postscript,
which takes up 31 pages. Referring to Newton's
Lemma I, Jurin says that in his former expres-
sion, the quantities "come nearer to equality
than to have any assignable difference between
them," it never was his intention to assert "that
during the time of the approach, the difference
between the quantities is not always assignable " ;
he meant "that, though they shall always have a
difference during the finite time, yet, before the
end of that time, their difference shall become
less than any quantity that can be assigned. And
if my words are taken in this sense, the Dr. 's
objection immediately falls to the ground " (p. (24)).
Mr. Jurin then gives a "demonstration" of the
following proposition: "If two lines (i) tend con-
stantly to equality with each other, (2) during any
finite time, as, for instance, an hour; (3) and thereby,
their difference become less than any quantity that
can be assigned, (4) before the end of the hour ;
then, at the end of that finite time, or at the end
of the hour, the lines will be equal." As to Dr.
Pemberton's charge that Jurin misinterprets Newton's

JURIN v. ROBINS AND PEMBERTON 135

nascent and evanescent increments, Jurin says that
he discussed this question with Robins. Newton's
words in the Quadrature* Curvarum, viz. finitarum
nascentium vel evanescentium, may mean "(i) finite
nascent or evanescent quantities, or (2) finite quan-
tities when they begin to be, or when they vanish.
But the former sense contradicts the second Lemma
of the second Book of the Principia, where Sir Isaac
Newton says, cave intellexeris particulas finitas . . .
and indeed it is contrary to the whole tenor of his
doctrine." The second interpretation is " perfectly
conformable to all the rest of Sir Isaac Newton's
works" (p. (32)). Jurin repeats that a nascent in-
crement is "an increment not yet arrived at any
assignable magnitude, how small soever." To Dr.
Pemberton's query, whether Jurin "has not here
attempted to define a non-entity," Jurin replies that
it "ought not to be called simply a non-entity, nor
simply an entity. It is a non-entity passing into
entity, or entity arising from non-entity, a begin-
ning entity, something arising out of nothing"

(P. (37))- '

141. The discussion is carried on from this time
in a journal called The Works of the Learned, into
which the Republick of Letters and another journal
had merged. In the February, 1737, issue Dr.
Pemberton appears with Some Observations on the
Appendix to the Present State of the Republick of
Letters for December, 1736, which enjoys the merit
of brevity, being limited to only two pages. Pem-
berton declares that in Newton's passage in the

136 LIMITS AND FLUXIONS

Quadratura Curvarum, <l Philalethes cannot remove
my objection by straining one or two of the words
to fit his sense " ; Newton meant there that vanish-
ing quantities should not be "otherwise than finite
quantities" (p. 157). Moreover, "what kind of
nothings they must be, which with any propriety can
be said to pass into somethings, and for this reason
can be capable of bearing proportions, before they are
become anything, certainly requires explanation."

A reply by Jurin in The Works of the Learned
for March, 1637, is kept within the very moderate
compass of 10 pages. The title of his contribution
is The Contents of Dr. Pembertorfs Observations pub-
lished the last month. Nothing here is of interest
in the interpretation of Newton.

Dr. Pemberton's reply in the April issue refers to
Jurin's phrase, "they come nearer to equality than
to have any assignable difference between them " :
" My objection to the interpretation of Philalethes
[in the Minute Mathematician , p. 88] is, that these
words, which compose the third article of that inter-
pretation, in conjunction with the fourth article can
have no other signification, than that the quantities
come nearer to equality than to have any difference
between them before that point of time, wherein
they are supposed by the second article to become
equal ; all which amounts to this inconsistency,
that there is a time, when the quantities have no
difference, and yet are not equal " (p. 306). Dr.
Pemberton again gives his endorsement of Robins's
interpretation of Newton.

fURIN v. ROBINS AND PEMBERTON 137

Jurin appears with a 1 2-page article in the May,
1737, number of The Works of the Learned, say-
ing : " He still ascribes to my words a meaning,
which I have again and again utterly disavowed ;
not only so, but he changes the words themselves,
putting any difference instead of any assignable differ-
ence" (p. 388). As to the Introduction to Newton's
Quadratura Curvarum, " in that very Introduction
Sir Isaac Newton has made use of infinitely little
quantities, in the sense I understand them, that is,
quantities which being at first finite, do by a gradual
diminution at last vanish into nothing and conse-
quently must, during their diminution, become less
than any quantity that can be assigned " (p. 389).
As to evanescent quantities being entities or non-
entities, "If this page were divided from top to
bottom into two equal parts, one black, and the
other white,1 and Dr. Pemberton were to ask me,
whether the middle line, which divides the two
parts, were black or white, I apprehend it would
be a direct answer to say, it is neither ; it cannot
properly be called either a black line, or a white
line ; it is the end of the white and beginning of
the black, or the end of the black and beginning of
the white" (p. 389). "I was apprised that Mr.
Robins had all along expressed the sentiments of
Dr. Pemberton " (p. 393). Dr. Pemberton still
refuses to give his interpretation of Newton's

1 As far as I know, Jurin is the first to use colour devices to illustrate
subtle points in evanescent quantity or in number. Jules Tannery, in
his Lemons d1 Algebre ct <? Analyse, Paris, 1906, p. 14, uses colour imagery
to illustrate the discussion of irrational numbers.

138 LIMITS AND FLUXIONS

Lemma. " Every body will be satisfied that the
true reason of his backwardness, is the fear he is
under, that I shall make good my promise, in shew-
ing, that his explanation is either a false one, or,
in case it be true, is to all intents and purposes the
very same with mine " (p. 396).

In June, 1737, Dr. Pemberton replies again, by re-
peating his previous assertion against Philalethes's
explanation of Newton's Lemma, given in the
Minute Mathematician, but does not permit him-
self to be drawn into giving an explanation of his
own of Newton's Lemma.

In Jurin's article in the July issue, 1737, we read :
"I did indeed take notice of the prudence Dr.
Pemberton used, in passing by my second inter-
pretation, which was so clear and plain, and was so
fully illustrated by examples, that there was no
possibility of perverting the sense of it" (p. 70).
"But since this dispute, which began upon matters
of science, . . . unless Dr. Pemberton shall see fit
to revive it by giving his so long demanded explica-
tion, I shall not judge it worth while to take notice
of what he may hereafter write."

Dr. Pemberton followed with some Observations in
the August, 1737, number, while in the September
number there appears " the last reply of Philalethes,"
and in the October number the final answer by
Pemberton. Thus ended a dispute which had for
some time ceased to contain much of scientific and
historic value.

JURIN v. ROBINS AND PEMBERTON 139

Debate over Robins 's Review of Treatises written by
Leonhard Euler, Robert Smith, and Jurin

142. Being in a somewhat combative mood,
Robins made attacks upon Euler's treatise on motion,
Dr. Robert Smith's optics, and Jurin's essay on vision.1

Robins's criticisms of Euler concern mainly the
philosophy of the Calculus. Robins quotes Euler's
third proposition, "That in any unequal motion
the least element of the space described may be
conceived to be passed over with an uniform motion,"
and then says, this "is not universally true," as,
for instance, "when those spaces are compared
together, which a body accelerated by any force
described in the beginning of its motion ; for the
ultimate proportion of the first of two contiguous
spaces, thus described in equal times to the second,
is not that of equality, but the ratio of i to 3,
as is well known to every one acquainted with
the common theory of falling bodies " (p. 2). In
another place (p. 4) Robins argues that the path
assigned by Euler to a certain body "is false even
on the confused principles of indivisibles." Some
passages in Robins involve the Leibnizian notation
in the calculus, and look quite odd in an eighteenth-
century publication prepared by a Briton in Great
Britain. Robins concludes that most of Euler's
errors "are owing to so strong an attachment to
the principles, he had imbibed under that inelegant

1 Remarks on Mr. Euler's Treatise of Motion, Dr. Smith's Compleat
System of Opticks, and Dr, Jurin s Essay upon Distinct and Indistinct
Vision. By Benjamin Robins, London, 1739.

LIMITS AND FLUXIONS

computist, who was his instructor, that he was
afraid to trust his own understanding even in cases,
where the maxims, he had learnt, seemed to him
contradictory to common sense " (p. 30). This
master was John Bernoulli.

143. Never losing an opportunity to engage in
controversy, Jurin wrote a treatise in reply.1 We
refer only to such parts of this pamphlet, and the
ones which followed it, as bear on fluxions or the
parties engaged in the discussions on fluxions.

In the preface Jurin says : l( I, it seems, am the
Reputed Author of the late dissertations under the
name of Philalethes Gantabrigiensis, and the other
Gentleman [Dr. Robert Smith] is ... suspected of
being my associate. ... If Dr. Smith were to tell
Mr. Robins, what he has often professed to other
persons, that he had no hand in those papers ; if to
confirm this he were to remind him, that Philalethes
has declared more than once, he wrote alone and
unassisted ; if I — But what signifies pleading, when
the execution is over ? Mr. Robins has already
vented his Resentment to the utmost. ..."

144. Not without interest is the following refer-
ence to young Euler in St. Petersburg, whose
scientific achievements have been so very extra-
ordinary. Jurin says that to make no reply to
Robins's criticisms " might be such a discouragement
to the hopeful young writer, whose name is prefixed

1 A Reply to Mr. Robins's Remarks on the Essay upon Distinct and
Indistinct Vision Published at the End of Dr. Smith's Compleat System
of Opticks. By James Jurin, M. D. , London, MDCCXXXIX.

JURIN v. ROBINS AND PEMBERTON 141

to their common labours, and who possibly, when
he comes to study suo Marte, and to see with his
own eyes, or to meet with abler instructors, may
make some figure in the Learned World, that pure
humanity induces me to oblige them with this one
Reply " (p. 54).

145. Of course, Robins wrote a tract in reply,1
but only the preface of this tract demands our
attention. In answer to the charge made by Jurin,
that he (Robins) had conducted the controversy
"with passion and abuse," Robins proceeds to
explain their past relations to each other.

(( About six years since a pamphlet was publish'd
under the title of the Analyst ; in which the author
endeavors to shew, that the doctrine of fluxions in-
vented by Sir Isaac Newton is founded on fallacious
suppositions. As that writer had a false idea of this
doctrine, ... I thought the most effectual method
of obviating his objections would be to explain . . .
what Sir Isaac Newton himself had delivered with
his usual brevity. . . . And with this view I pub-
lished a Discourse on Sir Isaac Newton's method
of fluxions, and of prime and ultimate ratios. But
in the mean time a controversy was carrying on
between the author of the Analyst and another,
who under the name of Philalethes Cantabrigiensis
had undertaken the defence of Sir Isaac Newton :
and as I at last perceived, both by the concessions

1 A Full Confutation of Dr. Jurirfs Reply to the Remarks on his
JEssay upon Distinct and Indistinct Vision. By Benjamin Robins,
London, 1740.

142 LIMITS AND FLUXIONS

of Philalethes, and the avowed opinions of others,
that the erroneous conceptions of the writer of trie
Analyst on this head were more prevalent even
amongst those, who approved of the method of
fluxions, than I had at first believed ; I thought, it
might be no unacceptable task more particularly to
shew those, who were thus misled, how irreconcile-
able their opinions were with the tenets of Sir Isaac
Newton, and how impossible it would be to defend
the accuracy of his doctrine on these their mistaken
suppositions ; and it was with this intention, that in
an account of my book inserted in the Present State
of the Republick of Letters, some of the errors con-
tained in the writings of Philalethes Cantabrigiensis
were endeavoured to be obviated.

" But tho' this discourse was written with great
caution, and only mentioned the principles objected
to without so much as naming or even insinuating
the treatises, from whence they were taken ; yet,
as Dr. Jurin, who was generally reputed the author
of them, was one, that I often conversed with ; at
my request, before this paper was printed, a common
friend carried to him the manuscript, and, without
pretending to suppose, whether he was, or was not
Philalethes, desired him to read it, and asked him
if he thought, Philalethes could be displeased with
any thing contained in it ; he was also told at the
same time, that if he believed any part of it could
give offence to that gentleman, whoever he were,
it should be struck out, or that 1 would even let
the whole design fall, if he desired it.

JURIN v. ROBINS AND PEMBERTON 143

" My friend brought me the Doctor's answer
importing, that he could not believe, my paper
would displease any one, since, if the tenets, I
excepted to, were really erroneous, it was reason-
able, they should be exposed ; and if otherwise, it
was the business of Philalethes to defend them . . .
it was however added, that I had in two places
censured doctrines, which, if I supposed them to
be the opinions of Philalethes, I must have mis-
apprehended him. Now ... I immediately ex-
punged them, and published the remaining part
in the Republick of Letters for October 1735, as
an account of my book on Sir Isaac Newton's
method of fluxions, and of prime and ultimate
ratios.

"To this Philalethes answered in the following
month, and I again replied, till five papers were
successively written in this controversy, that is,
three by me, and two by him. And all this time
so very desirous was I on my part of avoiding
irritating circumstances . . . that I thought even
the most intimate friend . . . could not be offended
with it. ... But alas . . . Philalethes in his reply,
part of which was published in the July following,
and the rest in the succeeding month, runs out into
the most extravagant heats of passion . . . charg-
ing me with dishonestly writing against the con-
victions of my own judgment. . . . After so gross
and unprovoked an abuse, ... I should surely have
been acquitted of any breach of decency, if ... I
had sharply exposed his ignorance in the subjects,

144 LIMITS AND FLUXIONS

he had attempted. But I chose, if possible, to
avoid the ridicule of quarreling on a matter of
mere speculation ... I again requested my friend
to speak to Dr. Jurin, and to represent to him the
inconveniencies, that would arise from the persever-
ance of Philalethes in his rash and groundless
calumny. My friend accordingly went to Dr. Jurin,
and carried with him an answer to so much of
Philalethes's paper, as was then published, and told
the Doctor, that he came to propose to him a method,
that might prevent the controversy betwixt me and
Philalethes from degenerating into a passionate
personal altercation . . . that therefore, if Dr. Jurin
thought it expedient, my paper should be given to
a certain gentleman, to whose impartiality and
knowledge of the subject in debate no exception
could be taken on either side ; and that if, when
that gentleman had perused it, he should believe,
I had in any instance changed my opinion from
my first entering into this dispute, I did then
promise to submit patiently and without reply to
any censures of unfairness and dishonesty, that
Philalethes . . . should hereafter think proper, . . .
[otherwise] it would then be but common justice,
that Philalethes should moderate the remaining
part of his performance. . . . But this proposal was
rejected. ... It was immediately given out, that
my friends had confessed me to have been foiled
in the argument ; and were now only sollicitous to
support me from the charge of unfairness. . . . The
reader will not wonder, if I resolved for the future

JURIN v. ROBINS AND PEMBERTON 145

to treat him with that freedom, which his unskilful-
ness authorised. ..."

146. The above preface constitutes what we may
call Robins's apologia pro vita sua. It seems only
fitting that Jurin should appear with a similar docu-
ment. This he did in a long Letter.^-

We make the following quotations from Jurin
(p. 8) :

" About five years ago some passages in a paper
of Mr. Robins, were shown to me . . . and a question
was put to me, whether I should take it ill, if those
passages were printed, it being intimated, that
Philalethes, against whom they were designed,
might possibly be some friend of mine : and indeed,
several persons were then guessed at, all of which
happened to be my friends. To this ... I gave
answer, that I should not at all take it ill. But I
added, that as I had read the controversy between
Philalethes and the Author of the Analyst, with some
attention, it seemed to me that in one or two
passages Mr. Robins imputed opinions to Philalethes,
which . . . that gentleman did not hold. . . . Also,
I took notice, that Mr. Robins did not rightly
explain Sir Isaac Newton's first Lemma. . . . But
when I desired to talk with Mr. Robins about the
Lemma, before the papers went to the press, as
imagining I could convince him that he was in the
wrong, answer was made, that the question was

1 A Lttter to . . . Esquire, In Answer to Mr. Robins's Full Confu-
tation of the Reply to his Remarks on the Essay upon distinct and
indistinct Vision. By James Jurin, M.D., London, 1741.

146 LIMITS AND FLUXIONS

not whether I thought him in the right or in the
wrong, but only whether I should take anything
amiss ; to which I replied as before. Upon talking
with another friend of Mr. Robins a day or two
after, I repeated my desire to talk with Mr. Robins
about his explanation of the Lemma, before his
papers went to the press : but was told that could
not be, for that the part of the papers where the
Lemma was spoke of, was to go to the press that
afternoon. ... I do not remember, that any offer
was made to me of * letting the whole design fall,
if I desired it.' Had any such offer been made, I
had at that time so much regard for Mr. Robins,
that I think I should at least have desired him to
stop the design, till he and I had examined the
Lemma together, in order to prevent his exposing
himself in the manner he has since done. As to
the second application made to me near a year
after, it may easily be judged, that I, who gave
these gentlemen no reason to think I had any in-
fluence over Philalethes, or so much as knew who
he was, could neither comply with nor reject their
proposal " (p. 9).

Remarks

147. The debate between Jurin and Robins is
the most thorough discussion of the theory of
limits carried on in England during the eighteenth
century. It constitutes a refinement of previous
conceptions.

Jurin possessed the more general conception of

JURIN v. ROBINS AND PEMBERTON 147

a limit in insisting that there are variables which
reach their limits. His interpretation of Newton
on this point appears to us more nearly correct than
that of Robins ; Jurin's geometric illustrations of
limit-reaching variable, intended to aid the imagina-
tion, though as he admits incapable of exhibiting the
process "all the way," are nevertheless interesting
(see our §§ 124, 132, 133). The imagination is
subject to limitations where the reason is still free
to act.

Robins, and after him Pemberton, deserve credit
in clearly, openly, and completely breaking away
from infinitely little quantities, and from prime and
ultimate ratios. Robins's conception of a limit was
narrow, but this narrowness had certain peda-
gogical advantages, since it did not involve a mode
of advance to the limit which altogether tran-
scended the power of the imagination to follow all
the way (see our §§ 117, 118, 129, 130).

It is interesting to observe that both Jurin and
Robins disavow belief in the possibility of a sub-
division of a line into parts so as to reach a point —
they assert "that such subdivision can never be
actually finished" (see our §§ 126, 132).

Robins discarded the use of Newton's moments
in developing the theory of fluxions (see our
§§ 119, 120).

Toward the end of his long debate with Robins,
Jurin begins to disavow infinitely small quantities.
He brings out the difference between infinitesimals
as variables, and infinitesimals as constants. He

H8 LIMITS AND FLUXIONS

rejects all quantity " fixed, determinate, invariable,
indivisible, less than any finite quantity whatsoever,"
but he usually admits somewhat hazily a quantity
"variable, divisible, that, by a constant diminution,
is conceived to become less than any finite quantity
whatever, and at last to vanish into nothing." (See
our §§ 132, 138, 141.)

While Berkeley's Analyst and Berkeley's replies
to Jurin and Walton involved purely destructive
criticism, the present controversy between Jurin and
Robins brought forth valuable constructive results.
Jurin's papers against Robins are decidedly superior
to those he wrote against Berkeley, though here too
they contained much that was not pertinent to the
subject and was intended merely to amuse the
general reader.

CHAPTER V

TEXT-BOOKS IMMEDIATELY FOLLOWING
BERKELEY'S ATTACK

148. The Analyst was published in 1734 ; two years
later appeared four books on fluxions. Thus, more
British text-books on this subject were published
in 1736 than in all the thirty years preceding. That
the Analyst controversy was largely the cause of
this increased productivity there can be no doubt.
We proceed to give an account of the books which
preceded the publication of Maclaurin's Treatise of
Fluxions, 1742.

John Colson > 1736

149. Newton's Method of Fluxions^ said to have
been written in 1671, was translated and first
published in 1736 by John Colson. Colson had
been a student at Christ Church, Oxford, which
he left without taking a degree. He was appointed

1 The Method of Fluxions and Infinite Series ; with its Application
to the Geometry of Curve- Lines. By the Inventor, Sir Isaac Newton , A?.,
Late President of the Royal Society. Translated from the Author's
Latin Original not yet made publick. To which is subjoined, A
Perpetual Comment upon the Whole Work, . . . By John Colson, M. A.
and F. R.S., Master of Sir Joseph Williamson's free Mathematical-
School at Rochester. London, M.DCC.XXXVI. This book was
reissued in 1758.

149

ISO LIMITS AND FLUXIONS

master of a new mathematical school founded at
Rochester, and, in 1739, Lucasian professor of
mathematics at Cambridge, in succession to Nicholas
Saunderson. Colson was a man of great industry
but only ordinary ability.

In his preface, Colson refers to the controversies
on fluxions, and says that the defenders as well as
their opponents were little acquainted with Newton's
own exposition, that this book now published for
the first time is "the only genuine and original
Fountain of this kind of knowledge. For what has
been elsewhere deliver'd by our Author, concerning
this Method, was only accidental and occasional"
(p. x). Colson accompanies Newton's book "with
an ample Commentary" and "particularly with an
Eye to the fore-mention'd Controversy " (p. x).
Colson in this preface represents Newton as hold-
ing the principle "that Quantity is infinitely
divisible, or that it may (mentally at least) so far
continually diminish, as at last, before it is totally
extinguish'd, to arrive at Quantities that may be
call'd vanishing Quantities, or which are infinitely
little, and less than any assignable Quantity. Or
it supposes that we may form a Notion, not indeed
of absolute, but of relative and comparative infinity "
(p. xi). Colson opposes " indivisibles," as also
the "infinitesimal method" and "infinitely little
Quantities and infinite orders and gradations of
these, not relatively but absolutely such " (p. xii).
He argues against " imaginary Systems of infinitely
great and infinitely little Quantities, and their

TEXT-BOOKS, 1736-1741 151

several orders and properties, which, to all sober
Inquirers into mathematical Truths, must certainly
appear very notional and visionary " (p. xii), for
" Absolute Infinity, as such, can hardly be the
object either of our Conceptions or Calculations,
but relative Infinity may, under a proper regula-
tion" (p. xii). Newton " observes this distinction
very strictly, and introduces none but infinitely
little Quantities that are relatively so." Colson
answers Berkeley's criticism in the Analyst of
Lemma 2, Book II, in the Principia in the follow-
ing manner : —

" Let X and Y be two variable Lines. . . . Let
there be three periods of time, at which X becomes
A-|tf, A,A + |#; and Y becomes B-|£, B, and
B + i£ . . . Then ... the Rectangle XY will
become . . . AB-J d&-\ £A + J ab, AB, and
AB + J0B + JM + \ab. Now in the interval from
the first period of time to the second ... its whole
Increment during that interval is |tfB + J£A — \ab.
And in the interval from the second period of time
to the third, ... its whole Increment during that
interval is | <zB + \ M + J ab. Add these two Incre-
ments together, and we shall have #B + $A for the
compleat Increment of the Product XY " (p. xiii),
called the ' ' Moment of the Rectangle " when a and
b are infinitely little.

Another mode of procedure is this: "the Fluxions
or Velocities of increase, are always proportional
to the contemporary Moments." " When the Incre-
ments become Moments, that is, when a and b are

152 LIMITS AND FLUXIONS

so far diminish'd, as to become infinitely less than
A and B ; at the same time ab will become infinitely
less than either a~B or £A (for 1 <zB . ab : : B . b, and
£A . ab : : A . a), and therefore it will vanish in
respect of them. In which case the Moment of the
Product or Rectangle will be #B + £A, as before"
(p. xv). Newton, however, prefers the more direct
way previously explained.

Proceeding to Newton himself, we find (on p. 24)
the following definition : " The Moments of flowing
Quantities (that is, their indefinitely small Parts,
by the accession of which, in infinitely small por-
tions of Time, they are continually increased) are
as the Velocities of their Flowing or Increasing.
Wherefore if the Moment of any one, as x> be repre-
sented by the Product of its Celerity x into an
indefinitely small Quantity o (that is, by x <?), the
Moments of the others v, y, zt will be represented
by vo, yo, zo ; because vo, xo, yo, and zo, are to each
other as v, x, y, and.?,'1 On p. 25 terms contain-
ing o as a factor "will be nothing in respect of the
rest. Therefore I reject them."

150. Colson appended extensive annotations to
Newton's treatise. In these annotations, p. 250,
Colson speaks of " smallest particles," but the term
" smallest " does not occur in Newton's definition.
However, Colson says that he does not mean
"atoms" nor "definite and determinate magni-
tude, as in the Method of Indivisibles," but things
"indefinitely small; or continually decreasing, till

1 Here aB . ab : : B . 6 means aB : ab : : B : 6.

TEXT-BOOKS, 1736-1741 153

they are less than any assignable quantities, and
yet may then retain all possible varieties of pro-
portion to one another. Becoming still more deeply
involved in the metaphysics of the subject, Colson
adds "that these Moments are not chimerical,
visionary, or merely imaginary things, but have an
existence sui generis, at least Mathematically and
in the Understanding, is a necessary consequence
from the infinite Divisibility of Quantity, which
I think hardly anybody now contests" (p. 251).
This he qualifies, "perhaps the ingenious Author
of ... The Analyst must be excepted, who is
pleased to ask, in his fifth Query, whether it be not
unnecessary, as well as absurd, to suppose that
finite Extension is infinitely divisible" (p. 251).
By ultimate ratio Colson means the ratio when the
arguments "become Moments " (p. 255). Fearing
that moments, infinitely little quantities, and the
like, "may furnish most matter of objection," he
says (p. 336) that the symbol o at first represents
a finite quantity, which then diminishes continually
till "it is quite exhausted, and terminates in mere
nothing." But "it cannot pass from being an
assignable quantity to nothing at once ; that were
to proceed per saltum, and not continually " ; hence
" it must be less than any assignable quantity what-
soever, that is, it must be a vanishing quantity.
Therefore the conception of a Moment, or vanishing
quantity, must be admitted as a rational Notion "
(p- 336)- Again: "The Impossibility of Concep-
tion may arise from the narrowness and imperfection

154 LIMITS AND FLUXIONS

of our Faculties, and not from any inconsistency in
the nature of the thing" ; these quantities tl escape
our imagination." Referring to imaginaries, a J — I
in the solution of cubic equations, Colson says
(pp. 338-9): "These impossible quantities . . .
are so far' from infecting or destroying the truth of
these Conclusions, that they are the necessary
means and helps of discovering it. And why may
we not conclude the same of that other species of
impossible quantities, if they must needs be thought
and call'd so ? . . . Therefore the admitting and
retaining these Quantities . . . 'tis enlarging the
number of general Principles and Methods, which
will always greatly contribute to the Advancement
of true Science. In short, it will enable us to make
a much greater progress and proficience, than we
otherwise can do, in cultivating and improving what
I have elsewhere call'd The Philosophy of Quantity. "
151. A review1 of this book contains the follow-
ing historical exposition. Sir Isaac Newton, 1665,
"found the Proportions of the Increments of inde-
terminate Quantities. These Increments or Aug-
menta Momentanea he called Moments, which others
called Particles, infinitely small Parts, and Indi-
visibles ; and the Velocities by which the Quantities
increased he called Motions, Velocities of Increase,
and Fluxions. He considered Quantities not as
composed of Indivisibles, but as generated by local
Motion, after the manner of the Ancients . . . and
represented such Moments [of Time] by the Letter o,

1 Republick of Letters, Art. XI, pp. 223-235, 1736.

TEXT-BOOKS, 1736-1741 iSS

or by any other Mark drawn into an Unit " (p. 228).
1 ' Fluxions are not Moments, but finite Quantities
of another kind." "When Mr. Newton is demon-
strating any Proposition, he considers the Moments
of Time in the Sense of the Vulgar, as indefinitely
small, but not infinitely so; and by that means
performs the whole work, in finite Figures, by the
Geometry of Euclid and Apollonius, exactly without
any Approximation : and when he has brought the
work to an Equation, and reduced the Equation to
the simplest Form, he supposes the Moments to
decrease and vanish ; and from the terms which
remain he deduces the Demonstration. But when
he is only investigating any Truth, or the Solution
of any Problem, he supposes the Moment of Time
to be infinitely little, in the Sense of Philosophers,
and works in Figures infinitely small."

James Hodgson, 1736

152. James Hodgson, a mathematical teacher and
writer, and a fellow of the Royal Society of London,
is the author of a book, The Doctrine of Fluxions.^

Hodgson says in his Introduction that "it is now
some years since the greatest Part of this Book was
prepared for the Press." There is no direct refer-
ence in the book to the Analyst controversy, but
the declaration is made that the principles upon
which fluxions rest need "fear no Opposition."

1 The Doctrine of Fluxions, founded on Sir Isaac Newton's Method,
Published by Himself in his Tract upon the Quadrature of Curves.
By James Hodgson, London, MDCCXXXVI.

156 LIMITS AND FLUXIONS

Hodgson also says in his Introduction that most
books on fluxions that have hitherto appeared
proceeded on the same principles as the Differential
Calculus, so that "by calling a Differential a
Fluxion, and a second Differential a second Fluxion,
etc. , they have . . . confusedly jumbled the Methods
together," although the principles are really "very
different." "The Differential Method teaches us
to consider Magnitudes as made up of an infinite
Number of very small constituent Parts put
together ; whereas the Fluxionary Method teaches
us to consider Magnitudes as generated by Motion
... ; so that to call a Differential a Fluxion, or a
Fluxion a Differential is an Abuse of Terms." In
the method of fluxions, "Quantities are rejected,
because they really vanish " ; in the differential
method they are rejected "because they are in-
finitely small." Hodgson adds that he always used
the differential method "'till I became acquainted
with the Fluxionary Method." He considers fluxions
of quantities (p. 50) "in the first Ratio of their
nascent Augments, or in the last ratio of their
evanescent Decrements," and gives an able and faithful
exposition of Newton's ideas as found in his
Quadrature of Curves. He cannot think "there is
any more difficulty in conceiving or forming an
adequate Notion of a nascent or evanescent Quantity,
than there is of a Mathematical Point " (p. xi). In
explaining the derivation of the fluxion of the
product xy = z he apparently permits (p. xv) the
small quantity o to " vanish," and thereupon divides

TEXT-BOOKS, 1736-1741 157

both sides of the equation xyo + yxo — zo by o.
However, in the exposition given on p. 50 he is
more careful and divides by o while o is an incre-
ment, and obtains yx+ xy + yzo = v. Then he says :
' 'Imagine the Quantity o to be infinitely dimin-
ished, or, which is the same thing, the Quantity
xy to return back again into its arising State ; then
the Quantity xyo, in this Case, into which o is multi-
plied, will vanish ; whence we shall have xy+yx=v
for the Fluxion of the Quantity proposed. " Hodgson
follows Newton closely and permits the variable to
reach its limit.

Thomas Bayes, 1736

153. An anonymous pamphlet of 50 pages, on
the Doctrine of Fluxions?- has been ascribed to Rev.
Thomas Bayes. This author contributed in 1763 to
the Philosophical Transactions a meritorious article
on the doctrine of chances.

The pamphlet of 1736 represents a careful effort
to present an unobjectionable foundation of fluxions.
"The fluxion of a flowing quantity is its rate or
swiftness of increase or decrease." Let a, b, x, and
y be flowing quantities, let A and B be permanent
quantities ; if a : b — K^x : B=Fj^, during any time T,
and at the end of that Time, a, b, x, y all vanish ;
then . . . the ratio of A to B is the last ratio of the
vanishing quantities a and b (p. 13). This definition
is "in effect the same" as that given by Newton.

1 Introduction to Doctrine of Fluxions and Defence of the Mathe-
maticians against . . . the Analyst, 1736.

158 LIMITS AND FLUXIONS

The author speaks of "that most accurate defini-
tion of the ultimate ratio's of vanishing quantities ;
which we have at the latter end of Sch. Lemma XI
Princip. [see our §§ 10-15], an<^ which is so plain,
that I wonder how our author [Berkeley] could help
understanding it ; which had he done, I am apt to
think that all his Analyst says concerning the pro-
portion of quantities vanishing with the quantities
themselves, had never been heard : For according
to this definition, we are not obliged to consider the
last ratio as ever subsisting between the vanishing
quantities themselves. But between other quantities
it may subsist, not only after the vanishing
quantities are quite destroyed, but before when
they are as large as you please. And the reason
why we consider quantities as decreasing continually
till they vanish, is not in order to make, but to find
out, this last ratio. Sir Isaac Newton does not
indeed say that this last ratio is the ratio with which
the quantities themselves vanish ; but whether he
herein speaks with the utmost propriety or not, is
a mere nicety on which nothing at all depends "
(p. 1 6, note).

Velocity "signifies the degree of quickness with
which a body changes its situation in respect to
space"; the fluxion of a quantity "signifies the
degree of quickness with which the quantity changes
its magnitude. " "And when our author asserts, that
in order to conceive of a second fluxion, we must
conceive of a velocity of velocity, and that this is
nonsense ; he plainly appeals to the sound and

TEXT-BOOKS, 1736-1741 159

not the sense of words ... if ... you make it
synonymous to the word Fluxion, then the velocity
of velocity ... is nothing but plain common
sense" (p. 19). Moments are not used by the
author. The author says that, were he to write
a treatise on fluxions, "in order to understand
equations where Fluxions of different orders are
jumbled together ; it would be convenient to re-
present all Fluxions not as before, but as quantities
of the same kind with their Fluents. . . . The
Fluxion of a quantity anyhow flowing at any given
instant is a quantity found out by taking it to the
Fluxion of an uniformly flowing quantity in the
ultimate proportion of those synchronal changes
which then vanish" (pp. 34, 35). The variables
x and xn have the synchronal augments o and

noxn~^-\^— LJ0V~2H-, etc., which are to one

another as i : nxn~* + ^n* ~ "' ox"-z+, etc. "Let

now these arguments vanish, and their last ratio
will be i : nxn~l." "This our author says is no
fair and conclusive reasoning, because when we
suppose the ' increments to vanish, we must suppose
their proportions, their expressions, and everything
else derived from the supposition of their existence
to vanish with them.' To this I answer, that our
author himself must needs know thus much, viz.
That the lesser the increment o is taken, the nearer
the proportion of the increments of x and xn will
arrive to that of I to nxtt~'ly and that by supposing

160 LIMITS AND FLUXIONS

the increment o continually to decrease, the ratio
of these synchronal increments may be made to
approach to it nearer than by any assignable
difference, and can never come up with it before
the time when the increments themselves vanish.
. . . For tho', strictly speaking, it should be
allowed that there is no last proportion of vanishing
quantities, yet on this account no fair and candid
reader would find fault with Sir Isaac Newton, for
he has so plainly described the proportion he calls
by this name, as sufficiently to distinguish it from
any other whatsoever : So that the amount of all
objections against the justice of this method in
finding out the last proportion of vanishing
quantities can arise to little more than this, that
he has no right to call the proportions he finds out
according to this method by that name, which sure
must be egregious trifling. However, as on this
head our author seems to talk with more than usual
confidence of the advantage he has over his oppo-
nents, and gives us what he says is the amount of
Sir Isaac's reasoning, in a truly ridiculous light, it
will be proper to see on whom the laugh ought to
fall, for I am sure somebody must here appear
strangely ridiculous, ... I readily allow whatever
consequence he is pleased to draw from it, if it
appears that Sir Isaac, in order to find the last
ratios proposed was obliged to make two incon-
sistent suppositions. To confute which nothing
more need be said than barely to relate the sup-
positions he did make.

TEXT-BOOKS, 1736-1741 161

"i. Then he supposes that x by increasing be-
comes x+o, and from hence he deduces the relation
of the increment of x and xn.

"2. Again, in order to find the last ratio of the
increments vanishing, he supposes o to decrease till
it vanishes, or becomes equal to nothing. . . .
These are evidently no more inconsistent and con-
tradictory, than to suppose a man should first go
up stairs, and then come down again. To suppose
the increment to be something and nothing at the
same time, is contradictory ; but to suppose them
first to exist, and then to vanish, is perfectly con-
sistent ; nor will the consequences drawn from the
supposition of their prior existence, if just, be any
ways affected by the supposition of their subsequent
vanishing, because the truth of the latter supposition
no ways would have been an inconsistency ; but to
suppose them first unequal, and afterwards to become
equal, has not the shadow of difficulty in it. ...
must confess there seems to be some objection against
considering quantities as generated from moments.
What moments, what \.\\z principia jamjam nascentia
finitarum quantitatum, are in themselves, I own, I
don't understand. I can't, I am sure, easily con-
ceive what a quantity is before it comes to be of
some bigness or other ; and therefore moments
considered as parts of the quantities whose moments
they are, or as really fixed and determinate quanti-
ties of any kind, are beyond my comprehension, nor
do I indeed think that Sir Isaac Newton himself did

thus consider them " (pp. 35-41).

1 62 LIMITS AND FLUXIONS

John Mutter, 1736

154. John Muller, a German by birth, dates his
Mathematical Treatise?* 1736, from the Tower of
London, and dedicates it to the master-general of
the ordnance. He was appointed in 1741 head-
master of the Royal Military Academy, Woolwich.
He was ' ' the scholastic father of all the great
engineers this country employed for forty years."

The author's method of explaining fluxions is
somewhat unique. " I make no use of infinitely
small quantities nor of nascent or evanescent
velocities ; and yet I think to have explained those
Principles, so that any Person of a moderate
capacity . . . may be fully convinced of the Truth
thereof" (Preface). He begins his conic sections
with the postulate: "Grant that two infinite
quantities, differing from each other by a finite
quantity, may be esteemed equal." He then
explains that this postulate "is here of use only
to shew the connection of the Conic-Sections," and
hastens to assure the reader that "whenever we
make use of it in the demonstration of any Proposi-
tion, we shall give always another Demonstration
independent on it."

In the Republick of Letters, June, 1736, occurs the
following comment :

" He introduces this [Conic-Sections] by a Postu-
latum that sounds very absurdly to those that are

1 A Mathematical Treatise : Containing a System of Conic- Sections ;
with the Doctrine of Fluxions and Fluents, Applied to various Subjects.
By John Muller. London, 1736.

TEXT-BOOKS, 1736-1741 163

not vers'd in mathematical Speculations. 'Grant,'
says he, ' that two infinite Quantities, differing from
one another by a finite Quantity, may be esteemed
equal.' Such would imagine that there could not
be two infinite Quantities ; or that if there could,
they must necessarily be absolutely and not only
reputedly equal. But however Hobbes or Berkeley
may talk of geometrical Fallacies, or these unex-
perienced People think, the Adepts in this Science
very well know, that more infinite Quantities than
two are possible, and that one Quantity may be in-
finitely greater than an infinite one, and yet be itself
infinitely less than a third. But enough of these
Ludibria Scientiae, that I may inform the Publick of
the more useful Theorems ..." (pp. 422, 423).

Muller considers in his text a curve generated
by a point * c urged by two powers acting in two
different directions, the one parallel to the Abscisses
and the other parallel to the Ordinates. I prove
from thence, that if this point (when arrived at a
given place) did continue to move with the velocity
it has there, it would proceed in a right line touch-
ing the Curve in that place ... So that the three
Directions being known in each place, the propor-
tion between the velocities of the urging powers is
likewise known." Fluxions are defined as velocities.
To find the fluxion of jj>2, he puts j/2=^r; the sub-
tangent of the parabola is 2y2 Since the subtangent
is to the ordinate as the velocities along the abscissa
and ordinate, he has 2^2 \y\ :x : yt or x—2yy^ and
.2yy is the required fluxion. Similarly, to find the

1 64 LIMITS AND FLUXIONS

fluxion of jF3, let^r=j3. Take« = ^, the u— x = z— y
xzz + zy+yy, or z— y : u—x=\ : zz + zy+yy. If
now y and z approach continually until they coincide
with an intermediate ordinate, then z—y and the
chord through the extremities of the ordinatesj and
z will likewise coincide with the tangent. Therefore,
the ordinate is to the subtangent as I is to ^yy.
Hence the proportion i : ^yy=y : x, or x= Wy* the
fluxion required. The same argument is applied to
ym. In these demonstrations appeal is made to a
geometric figure, and no attention is directed to the
ratio z—y :u—x for the difficult case when y = z.
The author remarks that ''though we commonly
say that . . . myym~l is the Fluxion of ym ; yet that
expression is not sufficiently accurate : Therefore,
the sense in which we desire to be understood is,
that I : mym~l : : y : myym~l, that is, unity is to
my"1"1, or y is to myym~v, as the fluxion or velocity
with which y is generated, is to the fluxion, or
contemporary velocity with which ym is generated,
and so for the rest " (p. 79). Thus, the emphasis
is placed upon the ratios of velocities.

Anonymous Translation x of Newton's
' ' Method of Fluxions , " 1737

154^. Colson's translation from the Latin ot
Newton's Method of Fluxions ', published in 1736,
was followed in 1737 by a second translation, which

1 A Treatise of the Method of Fluxions and Infinite Series •, With its
Application to the Geometry of Curve Lines. By Sir Isaac Newton, Kt.
Translated from the Latin Original not yet published. Designed by the
Author for the Use of Learners. London, MDCCXXXVII.

TEXT-BOOKS, 1736-1741 165

was anonymous. In it no mention is made of
Colson's edition. The anonymous translator says
in the preface : ' ' We have reason to believe that
what is here delivered, is wrought up to that Per-
fection in which Sir Isaac himself had once intended
to give it to the Publick. The ingenious Dr.
Pemberton has acquainted us that he had once pre-
vailed upon him to complete his Design and let
it come abroad. But as Sir Isaac's Death un-
happily put a stop to that Undertaking, I shall
esteem it none of the least Advantages of the
present Publication, if it may prove a means of
exciting that Honourable Gentleman, who is
possessed of his Papers, to think of communicating
them to some able Hand ; that so the Piece may at
last come out perfect and entire." As remarked
by G. J. Gray,1 the two translations were made
" from copies of the same manuscript," and differ
from each other only "in the mode of expressing
the work in English."

James Smith , 1737

155. In his New Treatise of Fluxions? Smith
says (Preface) : " What I call here the New Method,
and the Six Propositions immediately following,

1 A Bibliography of the Works of Sir Isaac Newton. By George
J. Gray. Second edition, Cambridge, 1907, p. 47.

2 A New Treatise of Fluxions \ containing, I. The Elements of
Fluxions , demonstrated in Two easy Propositions ', without Jirst or last
Ratios ; II. A Treatise of Nascent and Evanescent Quantities, first and
last Ratios, III. Sir Isaac Newton s Demonstration of the Fluxions
enlarged and illustrated: IV. Answers to the Principal Objections in
the Analyst. By James Smith, A.M., London, 1737.

1 66 LIMITS AND FLUXIONS

are entirely New . . . Our common Definition of
Motion, translatio corporis de loco in locum is
certainly imperfect, and I am inclined to think,
that Aristotle's old exploded Definition of Motion
will, some time or other, come into Vogue again.
Actus entis in potentia, quatcnus in potcntia est.
Motion is an Effect, and every Effect has a coin-
staneous Existence with the Action by which it is
produced. "

.The definitions with which Smith starts out are
not very reassuring. "The fluxion of a surface is
the Velocity of the generating Line." "The
velocity of a generating Line is the Sum of the
Velocities of all the Points of that Line, whether
these Points move with equal, or unequal Veloci-
ties." The rectangle xy "flows or increases by the
flowing of both its contiguous Sides " together ;
but it "flows into Length" by the velocity jiu-, and
"it flows into Breadth at the very same Instant
of Time" by the velocity xy. "Therefore the
Velocity with which it flows into Length and
Breadth is the Sum of the synchronic Velocities,"
xy+yx.

Nor is the second topic displayed with illumina-
tion. "A nascent Quantity is a Quantity in the
Instant of its commencing to exist." Similar to
this is the definition of " evanescent Quantity," as
are also the definitions of first and last ratios.
Interesting is the following proof that if "two
Quantities begin and cease to exist in any finite
Time T, . . . they have a first and a last Ratio,"

TEXT-BOOKS, 1736-1741 167

for, "if they have not a first Ratio, they have not
a second nor a third Ratio, etc. Therefore they
have no Ratio in the Time T ; but in the Time T
they are Quantities, " and ' c two quantities of the same
kind, as soon or so long as they have any Quantity,
Being or Existence [i.e. are not absolutely nothing],
have a Ratio the one to the other," that is, "they
have a Ratio and they have not a Ratio in the
Time T, which is absurd." Smith argues also that
since two quantities "cannot be in their first Ratio,
neither before nor after the Beginning of the Time
T, they must have been in their first Ratio at the
very Beginning of the Time T, just as they began
to exist." Near the close of this part of his book,
Smith reveals some of the subtleties of his topic
by stating an "Objection" and the "Answer" to
it. The Objection: "Nascent and evanescent
Quantities are Something or Nothing ; for, Inter ens
et non-ens non datur medium. If Something, then
the Ratio of evanescent Quantities is the same
with the Ratio before they were evanescent,
or when they had any finite Magnitude. . . „
If they are mere Nothing, or Non-quanta; then
B^/E£ = O/O = Q; . . . which is absurd." In the
"Answer" Smith says: "Evanescent Quantities
are really nothing, or Non-quanta ; for it is evident
. . . that upon £'s coinciding with B, and e's co-
inciding with E, the Increments B$ and E£ are
annihilated, and evanescent Quantities are never
accurately evanescent, but upon this or the like
Coincidence. And yet it does not follow that their.

1 68 LIMITS AND FLUXIONS

last Ratio, or the Ratio they nihilesce with, is
Nothing. For B£ / E£ is neither B& nor E£, nor
B£ and Ee, but a Mark or Expression of their
Ratio, which may be expressed as well by any
other Character. . . . The Increments are indeed
annihilated and gone, but their last Ratio remains,
and is as real as any Ratio they ever had"; . . .
they have as real a Ratio at the last Instant of
their Existence ; that is, when they are ceasing to
be Something, and commencing to be Nothing, as
they had at any instant preceding the last Instant
of their Existence." . . . "There is, sometimes,
something very strange in the Nature of these
evanescing Augments, and it is literally true of
them., what Juvenal figuratively says of Man.

— Mors solafatetur^
Quantula sunt hominum corpuscula —

We know nothing of them till they be dead and
gone. "

Of Part III, in which Smith "demonstrates"
Newton's Method of Fluxions, we quote only the
last sentence : (( I have made use of infinitely little
Quantities, and of a second Point as being next to
a first Point ; but this was only for Illustration sake.
There is not the least Occasion for any of these
Notions in the Demonstration."

In the last part of Smith's book, Berkeley's con-
tention, " No just Conclusion can be drawn from
two contrary Suppositions," is answered by the
statement, ' ' This is certainly true, in sensu com-
posite, but in sensu diviso is intirely false,"

TEXT-BOOKS, 1736-1741 169

We are tempted to make the remark that in
1737 Smith left the subject even more mysterious
than he found it.

Thomas Simpson, 1737

156. Thomas Simpson, the son of a weaver, was
a self-taught mathematician, and acquired a know-
ledge of fluxions through Stone's translation of De
L'Hospital's Analyse des infiniment petits. Simpson
was a mathematician of marked power, and influenced
considerably the teaching of mathematics in England.
In 1737 he brought out his New Treatise of Fluxions,1
which contains some novel features.

" The Fluxions of variable Quantities are always
measured by their Relation to each other ; and are '
ever expressed by the finite Spaces that would be -
uniformly described in equal Times, with the Veloci-
ties by which those Quantities are generated."

He finds it easy to show that the fluxion of a
rectangular area of constant height and uniformly
variable base is as the height drawn into the
velocity with which the base changes ; also that
the fluxion of a curvilinear area generated by an
abscissa moving with uniform velocity is at a given
point, as the ordinary y for this point, multiplied
by that velocity. This last result is applied to
finding the fluxion of xy.

Avoiding infinitely small quantities, Simpson finds

1 A New Treatise of Fluxions: -wherein the direct and inverse
Method are demonstrated after a new, clear and concise Manner, with
their Application to Physics and Astronomy. By Thomas Simpson,
London, 1737.

i;o LIMITS AND FLUXIONS

the ratio of the fluxions of x and x* thus : Let the
points ;;/ and n move so that the distance h de-
scribed by n shall always equal the square of the
distance g described by m in the same time. Then
(AR)2 = CS, (AR-Rr)2 = O, and jS = 2ARxRr-
(Rr)a. But jS is described with accelerated velocity
when m moves uniformly, hence ^S will be "less
than that which would be uniformly described in
the same time with the Velocity at the point S,
and greater than that which would be described

with the Velocity at the

r R

A. , . ^n point s ; and therefore

M mm m°

must be equal to the Dis-

P> f f ? «D tance that would be uni-
formly described with the
~9 Velocity at another point

e posited somewhere be-
tween S and s, in the same

Time that the other point m is moving over the
Distance rR ; therefore rR : 2ARxRr— (Rr)2 : :
g :£-(2AR — Rr), the Distance that would be de-
scribed with the Velocity of ny at the point et in the
same Time that m is moving over the Distance g :
Now therefore when the points r and s coincide
with R and S, then will e coincide with S ; . . . and
consequently (2AR — Rr)^ will then . . . become
2ARx^, equal to h the required Distance." The
critical part of this proof is ' * when the points r and s
coincide with R and S, then will e coincide with S. "
A modification of this proof is applied to xn.

Simpson's text marks a departure from Newton,

TEXT-BOOKS, 1736-1741 171

in the definition of fluxion. Newton makes it a
velocity, Simpson makes it a finite distance. On
\the necessity and wisdom of this change there can
(readily be difference of opinion. But there can be
no denying that Simpson developed his theory of
fluxions in a manner almost, though not entirely,
free from the objections against fluxions that had
been advanced by Berkeley ; infinitely small quan-
tities are nowhere used. A short but appreciative
review of this text appeared in The Works of the
Learned for July, 1737.

Benjamin Mar tin y 1739, 1759

157. Benjamin Martin was a mathematician, an
optical instrument maker, and a general compiler.
He was a self-educated man, and at one time taught
reading, writing, and arithmetic. His exposition of
fluxions, as found in his Elements of all Geometry 1
and in a later work, is below the standard usually
reached by him in mathematical writing.

This book, intended as an introduction to modern
mathematics, contains in an Appendix an epitome
of the doctrine of fluxions. " Since Fluxions are
the very small Increments and Decrements of the
Flowing Quantities, or the Velocities of the Motions
whereby they increase or decrease, 'tis plain that
those Fluxions, or Velocities, themselves may be
consider'd as Flowing Quantities, and their Fluxions
are call'd Second Fluxions ..." It would seem

1 nANFEftMETPIA ; or the Elements of all Geometry. By B, Martin,
London, M.DCC.XXXIX.

i;2 LIMITS AND FLUXIONS

that in this statement a fluxion is c ' very small " and
at the same time a t( velocity." A little later the
author refers to fluxions as "in the first Ratio of
Augmenta nascentia. " Evidently, in this Appendix,
covering twelve pages, the author has not succeeded
in presenting a consistent theory of fluxions.

A fuller exposition was given twenty years later
in the System of Mathematical Institutions, agreeable
to the Present State of the Newtonian Mathesis,
by Benjamin Martin, vol. i, London, MDCCLIX.
The theory is still confusing. " Indefinitely small
Spaces " (p. 362) are represented by x and y, which
are called the fluxions of x andj/, and said to repre-
sent the velocities of moving points. Newton is
reported to have at first delivered the idea of what
Martin calls a fluxion, under the name of momentum,
"a Term used in Mechanics to denote the Quantity
of Motion generated by a given Quantity of Matter
(A), and the Velocity (a) with which it moved con-
jointly. This Momentum therefore was properly
represented by (A#). . . . But instead of this
mechanical Notation, we now use xx and yy for the
Momenta, or Fluxions. ..." It is seldom that
one encounters a more grotesque conglomeration of
unrelated ideas than is presented here. Martin
gives John Rowe's mode of deriving the fluxions
of xy and xyz.

An Anonymous Text, 1741

158. An Explanation of Fluxions in a Short Essay
on the Theory. London: Printed for W. Innys, at the

TEXT-BOOKS, 1736-1741 173

West-End of St. Paul's, MDCCXLI. This anony-
mous publication of 16 pages was reprinted in 1809
in the fourth edition of John Rowe's Doctrine of
Fluxions ; it constitutes a real contribution to the
logic of fluxions. The pamphlet is offered "as an
Explanation of the Doctrine itself, and not of Sir
Isaac's Manner of delivering it." "About that,"
he says, * ' I don't mean, nor pretend to take a Part
in any Controversy." He defines fluxions thus:
'"The word Fluxion properly apply'd always sup-
poses the Generation of some Quantity (term'd
Fluent or Flowing Quantity) with an equable,
•accelerated, or retarded Velocity, and is itself the
Quantity which might be uniformly generated, in a
constant Portion of Time, with the Amount or
Remainder of that Velocity, at the Instant of find-
ing such Fluxion." "Hence, it will appear that
the first Fluxions of Quantities are as the Velocities
with which those Quantities are increas'd ; that
second Fluxions are as the Increase or Decrease of
such Velocities ; and that by second, third, fourth,
etc., Fluxions are meant Fluxions, whose Fluents
are themselves Fluxions to other proposed Quan-
tities ; and the manner of considering and determin-
ing them is the very same as tho' they were first
Fluxions, they being actually so to the Quantities
from which they are immediately derived " (p. 7).

Then follows the lemma :

"The Fluxion of the Area ABC, whether tri-
angular or curvilinear, is the Rectangle xy.n

Suppose B to move along AF while the ordinate

174

LIMITS AND FLUXIONS

y terminates in the curve AC ; " And, at any pro-
posed Position BC, conceive y to become constant,"'
while B "moves uniformly any constant Time, mn,
with the Velocity at B, over the Distance x or BD ;
for then will y in the Time mn uniformly generate
the Rectangle xyt which Rectangle is plainly the
Fluxion of ABC in this Position (per Defarit.)."
Then follows the illuminating scholium: "It has
been commonly objected to the Accuracy of Fluxions,
that the Trapezium or curvi-
linear Space BG&D, not the
Rectangle xyy is the Fluxion
geometrically exact. But,
this Objection is built, I
apprehend, upon a false Idea
of the Thing. It supposes a
Fluxion a complete Part of
a flowing Quantity, and an
Infinity of Fluxions to con-
stitute the flowing Quantity,,

which are Mistakes (per Definition and Lemma)
... if ,f be imagined infinitely little, an Infinity of
Increments may constitute the Area ABC. But, ini
Fluxions, our Reasoning is quite different: a Fluxion*
can no more be called a Part of the Fluent, than am
Effect a Part of the Cause. For Instance; from the:
Fluxion given we know the Fluent, and vice versa r
just as when a Cause is known to produce a certain
Effect, we can infer the one from a Knowledge of
the other. "

We shall find that later this reference to cause and

FIG. 8.

TEXT-BOOKS, 1736-1741 175

effect figured in a controversy carried on against
Simpson.

As regards the lemma given above, we shall see
that the same idea is elaborated in detail by
Maclaurin in his work and that a short and even
more convincing statement than the one given here
is found in the later, revised, text of John Rowe.

From the above lemma, the derivation of the
fluxion of xy becomes easy by considering the rect-
angle ABCG as made up of two parts AHCB and
AHCG, and applying the lemma to each part.

John Rowe, 1741, 1757, 1767

159. The first edition (1741) of John Rowe's
Doctrine of Fluxions 1 appeared anonymously. A
copy in the British Museum has the following added
by hand after the preface : " This is the first edition
of John Rowe's Fluxions. The second came out
with his name in 1757 with alterations and additions,
and the third came out in 1767 much improved."

In the first edition Rowe begins by stating his pro-
gramme: "To render the Doctrine of Fluxions plain
and easy" by explaining their nature "as deliver'd
both by Sir Isaac Newton and by Leibniz." Accord-
ing to Newton, " Fluxion is the same as velocity."

"Definition II [Foreigners Definition]. Quantities
are here suppos'd to be generated by a continual In-
crease, as before; and the indefinitely small Particles

1 An Introdttction to the Doctrine of Fluxions. Revised by several
Gentlemen well skilFd in the Mathematics. Felicibus inde Ingeniis
aperitur Iter—C/audt'an. London, M.DCC.XLT

LIMITS AND FLUXIONS

whereby they are continually increas'd, are calPd the
Fluxions of these Quantities " (p. 3). ' ' This is
the Notion of Fluxions as deliver'd by Leibnitz and
his Followers. But these Fluxions, we shall, in the
following Sheets, call by the Names of Moments,
Increments and Decrements ; that is, Moments or
Increments when the variable Quantities are increas-
ing, and Decrements when they are decreasing " (p. 4).

"As the Point b is
continually nearer to a
Coincidence with the
Tangent TBG the
nearer it approaches the
Point of Contact B ; so
if we conceive the Ordi-
nate cb to be moved on
till it concides with CB;
the very first moment
before its Coincidence, the Curve B^, and Right
line BG will be infinitely, or rather indefinitely
near a Coincidence with each other ; and conse-
quently, in that Case, the Increments Be, and eb
will come indefinitely near to measure the Ratio
of the Fluxions of the Absciss and Ordinate AC,
and CB, or the Velocities with which they flow at
the Point B . . . and therefore (because when any
Quantity is increas'd or decreas'd, but by only
an infinitely or indefinitely small Particle, that
Quantity may be consider'd as remaining the same
as it was before ;) these Increments may be taken
as Proportional to, or for the Fluxions in all Opera-

A C C

FIG. 9.

a F

TEXT-BOOKS, 1736-1741

177

tions ; and, on the contrary, the Fluxion for the
Increment " (pp. 5, 6). Accordingly, he deduces the
rules of operation by the use of increments, and in
the result substitutes the fluxion for the increment.

In rinding the fluxion of xy he lets x? and y' be
the increments, then the "increase in its nascent
state " is such that x'y' ' * bears no assignable Ratio
to either x'y or xyr (for as x'y' : x'y : \y' : y and y' by
Supposition is infinitely less than j," and can be
' ' expunged or rejected. "

1 60. The third edition (1767) was commented upon
by J. Stubbs, Fellow of Queen's College, Oxford,
as follows : "I received your valuable present, and

was much surprised to find
it so prodigiously improved.
Indeed, it so much resembles
a New Work, when compared
with the First Edition, that I
almost wish you had made
no mention of its being the
Third ; but left the two former
to be forgotten."

The .fluxion of xy is now
deduced thus: ''The fluxion
of the curvilinear space AEI is less than the fluxion of
the rectangle (of constant altitude) AH before EH
reaches BC, and greater after EH passes BC ; hence
at BC the two fluxions are alike and equal to yx.
Similarly, it follows that the rectangle AG (of con-
stant base) has the same fluxion xy at DB as has
the curvilinear space AFI. Hence the rectangle

1 B

/
/
/
/
/
/
/

\ f
FIG.

: (

10.

1 78 LIMITS AND FLUXIONS

of variable base and altitude AEIF with the vertex
I moving along the curve through B has the fluxion
xy+yx."

In a footnote Rowe expressed the belief that this
mode of deriving the rule is not open to criticism as
was the method of using increments which in 1 734 was
"smartly attacked by the late acute Dr. Berkeley."

Rowe proves by a geometrical method similar to
the above that the fluxion of a pyramid of fixed
vertex and slant edges, whose variable base xy
moves parallel to itself and whose variable altitude
is #, is xyz. Taking a parallelopipedon as equal to
three pyramids, he finds the fluxion of xyz to be
xyz + xyz -\-xyz. This new way of deriving the
fluxion of xyz was copied by "his friend" Benjamin
Martin in the Mathematical Institutions.

At the end of the third edition of Rowe's Fluxions
is a bibliography of English works on this subject,
and he ' ' particularly refers to the Works of his two
celebrated Friends, Mr. Emerson and the late Mr.
Simpson."

Berkeley Ten Years After

161. Berkeley, in his Sin's1 of 1744, expressed
himself as follows: "Concerning absolute space,
that phantom of the mechanic and geometrical
philosophers (§ 250), it may suffice to observe that
it is neither perceived by any sense, nor proved by
any reason, and was accordingly treated by the
greatest of the ancients as a thing merely visionary.

1 George Berkeley's Works, Edition by A. C. Fraser, vol. ii,
Oxford, 1871, p. 468 and note.

TEXT-BOOKS, 1736-1741 179

From the notion of absolute space springs that of
absolute motion ..." He continues in a foot;
note : " Our judgment in these matters is not to be
overborne by a presumed evidence of mathematical
notions and reasonings, since it is plain the mathe-
maticians of this age embrace obscure notions, and
uncertain opinions, and are puzzled about them,
contradicting each other and disputing like other
men : witness their doctrine of Fluxions, about
which, within these ten years, I have seen published
about twenty tracts and dissertations, whose authors
being utterly at variance, and inconsistent with each
other, instruct by-standers what to think of their
pretensions to evidence."

Remarks

162. In these publications no reference is made
to the Jurin-Robins controversy, though Berkeley's
Analyst is frequently discussed. Excepting only in
i Benjamin Martin, the definition of a fluxion as a
"differential" nowhere appears. Therein we see
a step in advance.

The influence of Newton's Quadrature of Curves
(1704) is evident almost everywhere. An improve-
ment in the mode of deriving the fluxion of a
' ' product " appears in the anonymous Explanation of
Fluxions and in the revised text by John Rowe
(our §§ 158, 1 60).

Noteworthy is Thomas Simpson's new definition
of fluxions ; this new definition plays an important
role during the rest of the century.

i8o LIMITS AND FLUXIONS

163. We quote Sir William Rowan Hamilton's
remarks on the lemma of the anonymous Explana-
tion of Fluxions (1741) and the derivation of the
fluxion of xy, based upon it. Hamilton knew this
proof as it is given in a later edition of Simpson's
fluxions. Says Hamilton i1 " I notice that Thomas
Simpson treats fluxions as finite . . . Thomas
Simpson's conceptions appear to have been very
clear and distinct, and I do not venture to say that
the geometrical investigation which he gives of the
fluxion of a rectangle, avowedly supplied to him by
a young but unnamed friend, is insufficient in itself,
but it fails to convince me, perhaps because I was
not early accustomed to fluxions. Certainly there
is no neglecting of ab, or xy, as small; for in fact
that rectangle of the fluxions is not represented at
all in his Figure ... He conceives the varying
rectangle xy to be the sum of two mixtilinear triangles ,
of which the two separate fluxions are yx and xy.
This is very ingenious, but I do not feel sure to
what degree I could rely on it and build upon it any
superstructure, if I were now coming, for the first
time, as a learner, to the subject. However, I
suppose that a pupil, if reasonably modest or even
prudent, will take, for a while, his teacher's state-
ments upon trust ; reserving to himself to return
upon them, and to examine closely their truth and
logic when he shall have acquired some degree of
familiarity with the subject taught."

1 Life of Sir William Rowan Hamilton, by R. P. Graves, vol. iii, p. 571.

CHAPTER VI
MACLAURIN'S TREATISE OF FLUXIONS, 1742

164. Colin Maclaurin was educated at the Uni-
versity of Glasgow, and through the influence of
Newton was elected professor at the University of
Edinburgh. Maclaurin's book on fluxions has been
considered the ablest and most rigorous text of
the eighteenth century. It was pronounced by
Lagrange ' ' le chef d'ceuvre de geometric qu'on
peut comparer a tout ce qu' Archimede nous a
laisse" de plus beau et de plus ingenieux. " *

In the preface to his Treatise of Fluxions 2
Maclaurin says: "A Letter published in the Year
1734, under the Title of the Analyst, first gave
Occasion to the ensuing Treatise. ... In the
mean Time the Defence of the Method of Fluxions,
and of the great Inventor, was not neglected.
Besides an Answer to the Analyst that appeared
very early under the Name of Philalethes Canta-
brigiensis ... a second by the same Hand in
Defence of the first, a Discourse by Mr. Robins, a

1 Mini, de FAcad. de Berlin, 1773 ; quoted in the art. " Maclaurin "
in Sidney Lee's Diet, of National Biography.

2 A Treatise of Fluxions in Two Books. By Colin MacLaurin, A.M.,
Professor of Mathematics in the University of Edinburgh, and Fellow
of the Royal Society. Edinburgh, MDCCXLU.

1 82 LIMITS AND FLUXIONS

Treatise of Sir Isaac Newton's with a Commentary
by Mr. Colson, and several other Pieces were
published on this Subject. After I saw that so
much had been written upon it to so good Purpose ;
I was the rather induced to delay the Publication of
this Treatise, till I could finish my Design. . . .
The greatest Part of the first Book was printed in
1737 ; But it could not have been so useful to the
Reader without the second. ... In explaining the
Notion of a Fluxion, 1 have followed Sir Isaac
Newton in the first Book . . . ; nor do I think
that I have departed from his Sense in the second
Book ; and in both I have endeavoured to avoid
several Expressions, which, though convenient,
might be liable to Exceptions, and, perhaps, occasion
Disputes. I have always represented Fluxions of
| all Orders by finite Quantities, the Supposition of
an infinitely little Magnitude being too bold a
Postulatum for such a Science as Geometry. But,
because the Method of Infinitesimals is much in
use, and is valued for its Conciseness, I thought it
was requisite to account explicitly for the Truth,
and perfect Accuracy, of the Conclusions that are
derived from it . . . "

165. In the Introduction to his Fluxions Maclaurin
says: ". . . When the certainty of any part of
geometry is brought into question, the most effectual
way to set the truth in a full light, and to prevent
disputes, is to deduce it from axioms or first prin-
ciples of unexceptionable evidence, by demonstra-
tions of the strictest kind, after the manner of the

\Toface page 182

COXI19 MA<

MACLAURIN'S TREATISE,' 1742 183

antient geometricians. This is our design in the
following treatise ; wherein we do not propose to
alter Sir Isaac Newton's notion of a fluxion, but to
explain and demonstrate his method, by deducing
it at length from a few self-evident truths, in that
strict manner : and, in treating of it, to abstract
from all principles and postulates that may require
the imagining any other quantities but such as may
be easily conceived to have a real existence. We
shall not consider any part of space or time as
indivisible, or infinitely little ; but we shall consider
a point as a term or limit of a line, and a moment
as a term or limit of time ... [p. 41]. If we are
able to join infinity to any supposed idea of a deter-
minate quantity, and to reason concerning magni-
tude actually infinite, it is not surely with that
perspicuity that is required in geometry. In the
same manner, no magnitude can be conceived so
small, but a less than it may be supposed ; but
we are not therefore able to conceive a quantity
infinitely small . . ."

1 66. In the posthumous work, An Account of Sir
Isaac Newton's Philosophical Discoveries, by Colin
Maclaurin, 2nd ed. , London, 1750, there is printed
a life of Maclaurin, from which we glean the follow-
ing (pp. viii, ix, and xviii) relating to Berkeley's
attack in the Analyst :

"Mr. Maclaurin found it necessary to vindicate
his favourite study, and repel an accusation in which
he was most unjustly included. He began an
answer to the bishop's book ; but as he proceeded,

1 84 LIMITS AND FLUXIONS

so many discoveries, so many new theories and
problems occurred to him, that, instead of a vindi-
catory pamphlet, his work came out a complete
system of fluxions, with their application to the
most considerable problems in geometry and natural
philosophy. This work was published at Edinburgh
in 1742. . . . His demonstrations had been, several
years before, communicated to Dr. Berkeley, and
Mr. Maclaurin had treated him with the greatest
personal respect and civility : notwithstanding
which, in his pamphlet on tar-water,1 he renews
the charge, as if nothing had been done ; for this ex-
cellent reason, that different persons had conceived
and expressed the same thing in different ways. . . .
r Mr. Maclaurin found it necessary, in demonstrating
the principles of fluxions, to reject altogether those
exceptionable terms [infinite and infinitesimal}, and
to suppose no other than finite determinable quan-
tities, such as Euclid treats of in his geometry."

167. In Chapter I, p. 57, Maclaurin defines a
fluxion : " The velocity with which a quantity flows,
at any term of the time while it is supposed to be
generated, is called its Fluxion which is therefore
always measured by the increment or decrement
that would be generated in a given time by this
motion, if it was continued uniformly from that
term without any acceleration or retardation : or
it may be measured by the quantity that is gener-
ated in a given time by an uniform motion which
is equal to the generating motion at that term."

1 In the second edition Berkeley gave the article the name of Sin's,

MACLAURIN'S TREATISE, 1742 185

The term velocity had been under dispute, par-
ticularly in the controversy between Berkeley and
Walton. Maclaurin evidently perceived the diffi-
culty in arguing that variable velocity is a physical
fact; he says (p. 55), "the velocity of a variable
motion at any given term of time is not to be?
measured by the space that is actually described
after that term in a given time, but by the space
that would have been described if the motion had
continued uniformly from that term. If the action
of a variable power, or the velocity of a variable
motion, may not be measured in this manner, they
must not be susceptible of any mensuration at all " —
an argument not likely to convince or silence hostile
critics. He quotes Barrow's definition of velocity —
' ' the power by which a certain space may be
described in a certain time." In discussing ' 'power"
Maclaurin brings in the consideration of ' ' cause "
and "effect" in a way that sounds- odd in a work
laying the foundations to the abstract doctrine of
fluxions. Maclaurin uses the word " limit," without
giving it a formal definition. Theorem XII reads :
"The velocity of a motion that is accelerated or
retarded perpetually, is, at any term of the time,
to the velocity of an uniform motion, in a ratio that
is always a limit between the ratio of the spaces
described by these motions in any equal times
before that term, and the ratio of the spaces de-
scribed by them in any equal times after it."

1 68. In the Philosophical Transactions, vol. xlii,
for the years 1742-43, London, 1744, Maclaurin

1 86 LIMITS AND FLUXIONS

gives an account of his Treatise of Fluxions. On
p. 330 of these Transactions it is pointed out that
''the Theory of Motion is rendered applicable to
this Doctrine with the greatest Evidence, without
supposing Quantities infinitely little or having
recourse to prime or ultimate Ratios." Again
(p. 336): "There is, however, no Necessity for
considering Magnitude as made up of an infinite
Number of small Parts ; it is sufficient, that no
Quantity can be supposed to be so small, but it
< may be conceived to be diminished further ; ,and it
is obvious, that we are not to estimate the Number
of Parts that may be conceived in a given Magni-
tude, by those which in particular determinate
Circumstances may be actually perceived in it by
Sense ; since a greater Number of Parts become
visible in it by varying the Circumstances in which
it is perceived." Of importance is the following
(p. 336): "We shall therefore observe only, that
after giving some plain and obvious Instances,
wherein a Quantity is always increasing, and yet
never amounts to a certain finite Magnitude (as,
while the Tangent increases the Arc increases but
never amounts to a Quadrant)." That a variable
need not reach its limit is also emphasised in other
passages, as for instance (pp. 337, 338): "In like
manner a curvilineal Area . . . may increase, while
the base is produced, and approach continually to a
certain finite Space, but never amount to it. ...
A Spiral may in like manner approach to a Point
continually, and yet in any Number of Revolutions

MACLAURIN'S TREATISE, 1742 187

never arrive at it. ... The Author insists on these
Subjects, the rather that they are commonly described
in very mysterious Terms, and have the most fertile
of Paradoxes of any Parts of the higher Geometry."

The ideal of mathematical rigour, as entertained
by eighteenth-century writers, was reached by the
Greek geometricians, Euclid and Archimedes. To
derive the rules of fluxions by the rigorous methods
of the ancients was the ambition of Maclaurin.
Barring some obvious slips that are easily remedied,
Maclaurin certainly reached the ideal he had set.
Nor is this so very strange. Fluxions involve
questions concerning limits ; the ancients overcame
the difficulties of such questions by their method of
exhaustion. It was a rigorous method, but dread-
fully tedious. Maclaurin secured his aim at a
tremendous sacrifice. His work on fluxions consists
of 763 good-sized pages ; the first 590 pages do not
contain the notation of fluxions at all ; they deal
with the derivation of the fluxions of different
geometric figures, of logarithms, of trigonometric
functions, also with the discussions of maxima and
minima, asymptotes, curvature, and mechanics, in
a manner that the ancients might have followed,
and with the verbosity of which the ancients are
guilty. The consequence was that the work was
not attractive reading.

Maclaurin was fully aware of the value of a good
notation and ease of operation, for he says of the
doctrine (p. 575): "The improvements that have
been made by it, either in geometry or in philo-

1 88 LIMITS AND FLUXIONS

sophy, are in a great measure owing to the facility,
conciseness and great extent of the method of com-
putation, or algebraic part. It is for the sake of
these advantages that so many symbols are employed
in algebra." But to Maclaurin it seemed "worth
while to demonstrate the chief propositions of this
method in as clear and compleat a manner as
possible, if by this means we can preserve this
science from disputes " (p. 102). We shall see that
Maclaurin's book did not stop disputes. Had the
book been read more, it might have been more
effective in this respect. Our studies have led us
to the conclusion that Maclaurin was not widely
read. A second edition of his Fluxions did not
appear until 1801. His work was praised highly,
but seldom used and digested. We might say of
Maclaurin what has been said of the German poet
Klopstock : —

"Wer wird nicht einen Klopstock loben?
Doch wird ihn jeder lesen ? — Nein.
Wir wollen weniger erhoben,
Und fleissiger gelesen seyn."

Remarks 1

169. To what extent, if any, Maclaurin may have
been influenced by Robins in the mode of treating

1 In 1745 there appeared an anonymous publication on fluxions
which we have not had the opportunity to examine ; it was entitled,
The Harmony of the Ancient and Modern Geometry asserted. In
A. C. Eraser's edition of Berkeley's Works, vol. iii, Oxford, 1871,
p. 301, it is referred to as follows: "This last and forgotten tract
consists of papers given in to the Royal Society in 1742, and treats
fluxions as a particular branch of an alleged more general reasoning,
called the doctrine of maximinority and imnimajority."

MACLAURIN'S TREATISE, 1742 189

fluxions it is difficult to say. Later we shall see
what James Wilson states on this point. Certain
it is that Maclaurin's views agree much more closely
with those of Robins than those of Jurin. Maclaurin
stood for the narrower view of limits — limits none
of which are reached by the variable. However,
the conception of limits does not receive as great a
degree of emphasis with Maclaurin as it does in the
Discourse of Robins.

Of Maclaurin's Fluxions, Professor Kelland has
remarked: "The Analyst did good service to
science, if in no other way, at least by giving
occasion to this last work. The principles of the
method had been previously exhibited in a concise
and obscure manner ; Maclaurin developed them
after the manner of ancient geometers."

In 1749, Maclaurin's Treatise of Fluxions was
translated into French by Esprit Pezenas, director
of the observatory at Avignon.

As we look back, we see that the eight years im-
mediately following Berkeley's Analyst were eight
great years, during which Jurin, and especially
Robins and Maclaurin, made wonderful progress in
the banishment of infinitely small quantities and the (
development of the concept of a limit. Both before
and after that eight-year period there were published
books in Great Britain containing a mixture of Con-
tinental and British conceptions of the new calculus,
a superposition of British symbols and phraseology
upon the older Continental concepts.

CHAPTER VII

TEXT-BOOKS OF THE MIDDLE OF
THE CENTURY

John Stewart, 1745

170. John Stewart, professor of mathematics at
Marischal College in Aberdeen, is known as the
translator into English, with commentaries, of
Newton's Quadrature of Curves and Analysis by
Equations of an Infinite Number of Terms. *

The translator spares no pains in the endeavour
to remove any obscurities which the ordinary reader
might encounter therein. Newton's Quadrature of
Curves takes up 33 pages in John Stewart's volume;
Stewart's explanations thereof fill 287 pages. Re-
ferring to the controversy between Berkeley and
Jurin, Stewart says that "because the Doctrine of
prime and ultimate Ratios has been so much con-
troverted of late, I shall here enquire whether
we have any distinct Idea thereof." He quotes
Newton's Lemma I in Book I, Section I of the
Principia, also the proof of it, and then argues that
the limit is reached, for "a Difference less than

1 Sir Isaac Newton's Two Treatises of the Quadrature of Curves, and
Analysis by Equations of an Infinite Number of Terms, explained.
By John Stewart. London, 1745.

190

TEXT-BOOKS OF MIDDLE OFCENTUR Y 191

any Thing assignable, is the same Thing as no
Difference at all : for repeat it as often as you
please, it can never be equal to any finite Quantity :
and therefore can bear no Ratio to it, by Def. 4,
Bk. 5 [of Euclid's] Elements" (p. 37). Stewart
gives definitions of ultimate ratio of quantities and
of evanescent quantities, also definitions of prime
ratio of quantities and of nascent quantities. The
following is a specimen: "The ultimate Ratio of
evanescent Quantities is the Limit to which the
Ratio of variable Quantities diminishing without
Bound, continually approaches, to come nearer to
it than by any given Difference ; but which never
goes beyond ; yet no sooner attains to, than the
Quantities being diminished infinitely, vanish." The
following additional statement follows closely the
language of Newton (p. 39): "If any one should
object that there can be no ultimate Ratio of
continually diminishing and at last evanescent
Quantities : because before they vanish it is not the
last ; and after they vanish, they have no Ratio.
The Answer is, that the ultimate Ratio is neither
the Ratio of them before they vanish ; nor after
they vanish ; but the Ratio wherewith they vanish,
or the Limit to which their varying Ratio no sooner
arrives, than they vanish ; . . . that Ratio they
have that very Instant they vanish. ... It signi-
fies nothing to say ultimate Quantities cannot be
assigned, in regard Quantity is divisible without
End : for it is not the Quantities themselves that
are hereby determined, but only their Ratio :

192 LIMITS AND FLUXIONS

which is capable of being determined." This
matter, says Stewart, has been so clearly explained
by Newton, "that the great Dust which has been
raised of late about the Whole of this Doctrine,
must be owing to Weakness, or some worse
Principle " (p. 40).

William Emerson, 1743 (?), 1757, 1768

171. William Emerson was a self-taught mathe-
matician ; he wrote many mathematical texts which
indicate a good grasp of existing knowledge, but not
great originality. His Doctrine of Fluxions appeared
at London in 1743 (?). We have before us the third
edition, 1768. From it we quote as follows :

"The Velocity of the Increase of any generated
Quantity, or the Degree of Quickness (or Slowness)
wherewith the new Parts of it continually arise, is
called its Fluxion. "

"The indefinitely small Portions of the Fluent
which are generated in any indefinitely small Por-
tions of Time are called Moments or Increments . "

". . . The Moments and Fluxions ought not to
be confounded together, since the Moments (being
generated by Fluxions) are as different from the
Fluxions, as any Effect is different from its Cause."

The following is given as an axiom :

" Quantities, which in any finite Time continually
converge to Equality, and before the End of that
k^) Time, approach nearer to one another than by any
^k given Difference, do at last become equal."

" If any should think this not clear enough to

TEXT-BOOKS OF MIDDLEOF CENTUR Y 193

pass for an Axiom, he may consider it thus ; let
D be their ultimate Difference, therefore they
cannot approach nearer to equality, than by that
given Difference D, contrary to the Hypothesis ;
which Supposition is absurd in all Cases except
when D is nothing."

To find the fluxion of bxmym, he lets ox, oy be
moments, expands the powers of x+ox andjj/ + 0j>,
and finds the increments. Then he divides * ' by
the indefinite Quantity o." "But since the
(Velocity or) Fluxion is required wherewith that
Moment first arises, in this Case the Moments ox
and oy will also be just arising and therefore
nothing, and consequently o will be nothing, and
therefore all the Terms wherein it is found will be
nothing." The final result then follows. In his
Preface Emerson claims that * * Velocity must be
looked upon as the proper efficient Cause of the
Space described ; and the Space described the
adequate Effect of that Cause. " . . . "No incre-
ment can be taken so small, but it is still further
divisible ad infinitum ; and since the Velocity is
by Supposition continually variable, it is plain,
there can be no two Points of the Increment in
both of which the Velocity is accurately the same.
It is therefore most manifest, that the Velocity here
enquired after is peculiar to one only indivisible
Point ; . . . that the Velocity in any given Point
of the Line described . . . has a certain, fixed,
determinate Value. . . . Here a metaphysical
Disputant may demand, how it comes to pass, that

194 LIMITS AND FLUXIONS

any Velocity which continues for no Time at all, can
possibly describe any Space at all ; or whether its
Effect be absolutely nothing, or an infinitely small
Quantity, or what it is. Here then it is, that our
Reason is at a Stand, and the human Faculties are
quite confounded, lost, and bewildered. . . . Now
whether such subtile Questions will be ever de-
termined, or not, yet there is one Refuge for us,
viz. that it is nothing at all to our Purpose what
they are : . . . The Method of Fluxions has
no Dependence on these mysterious Disquisitions.
What I apprehend the Method of Fluxions to be
concerned in, is ... what a ... variable Velocity
can produce in the whole. And here I think no
Reason can be assigned, why a variable Cause
should not produce a variable Effect, . . . though
we have no Ideas at all of the perpetually arising
Increments, or their Magnitude in their nascent or
evanescent State, that have so much, and to so little
Purpose, confounded and puzzled the mathematical
World."

Thomas Simpson, 1750

172. Simpson's Treatise of Fluxions of 1737 has
already been noticed (our § 1 56). His text of
1750, The Doctrine and Application of Fluxions ,
London, is new, not only in the title, but to some
extent also in the mode of exposition. He says in
his Preface (1750) that he has used a tract entitled
An Explanation of Fluxions in a Short Essay on the
Theory, printed by W. Innys and written by one of

TEXT-BOOKS OF MIDDLE OF CENTUR Y 195

his friends who was too modest to put his name to
it. (See our §§ 158, 160, 163.) Simpson used his
friend's manner of determining the fluxion of a rect-
angle and of illustrating fluxions of higher order.
Simpson defines a fluxion as follows :

" The Magnitude by which any Flowing Quantity
would be uniformly increased, in a given Portion of
Time, with the generating Celerity at any proposed
Position, or Instant (was it from thence to continue
invariable), is the Fluxion of the said Quantity at
that Position, or Instant."

The derivation of the fluxion of xy is explained
after the manner adopted by John Rowe, both
..authors being indebted for it to the author of An
Explanation of Fluxions in a Short Essay on the
Theory. The same definitions and explanations of
the fundamentals are given by Thomas Simpson
in the last part of his Select Exercises for Young
Proficients in the Mathematicks, 1752. In the
preface to his Fluxions of 1750, Simpson touches
some points of philosophic interest. He says :

"By taking Fluxions as meer Velocities ', the
Imagination is confin'd, as it were, to a Point, and
without proper Care insensibly involv'd in meta-
physical Difficulties : But according to our Method
of conceiving and explaining the Matter, less
Caution in the Learner is necessary, and the higher
Orders of Fluxions are render'd much more easy
and intelligible — Besides, tho' Sir Isaac Newton
defines Fluxions to be the Velocities of Motions^
yet he hath Recourse to the Increments, or

196 LIMITS AND FLUXIONS

Moments, generated in equal Particles of Time, in
order to determine those Velocities ; which he after-
wards teaches us to expound by finite Magnitudes
of other Kinds : Without which (as is already hinted
above) we could have but very obscure Ideas of
higher Orders of Fluxions : For if Motion in (or at)
a Point be so difficult to conceive, that Some have,
even, gone so far as to dispute the very Existence
of Motion, how much more perplexing must it be
to form a Conception, not only, of the Velocity of a
Motion, but also infinite Changes and Affections of
It, in one and the same Point, where all the Orders
of Fluxions are to be considered.

"Seeing the Notion of a Fluxion, according to
our Manner of defining It, supposes an Uniform
Motion, it may, perhaps, seem a Matter of Diffi-
culty, at first View, how the Fluxions of Quantities,
generated by Means of accelerated and retarded
Motions, can be rightly assigned ; since not any,
the least, Time can be taken during which the
generating Celerity continues the same : Here,
indeed, we cannot express the Fluxion by any
Increment or Space, actually generated in a given
Time (as in uniform Motion). But, then, we can
easily determine, what the contemporary Increment,
or generated Space would be , if the Acceleration, or
Retardation, was to cease at the proposed Position
in which the Fluxion is to be found : Whence the
true Fluxion, itself, will be obtained, without the
Assistance of infinitely small Quantities, or any
metaphysical Considerations,"

TEXT-BOOKS OF MIDDLE OFCENTUR Y 197

N id 10 las Sounder son ^ 1756

173. At the age of twelve months Saunderson
lost his eyesight by small-pox ; nevertheless, he
rose to prominence. He studied at Christ's College,
Cambridge, and in 1711 succeeded Whiston as
Lucasian professor of mathematics at Cambridge.
His Fluxions 1 is a posthumous work.

We read (p. i) : " Let AB represent any
Moment of Time, whether finite or infinitely small
it matters not, terminated by the two Instants
A and B. Let x be the Value of any flowing or
growing Quantity at any Instant A, whose Velocity
at that Instant is such, that if it was to flow during
the whole Moment AB with this Velocity, it would
gain a certain Increment represented by x ; then is
this Quantity x called the Fluxion of x at the
Instant A, when the Value of the flowing Quantity
was x." In the scholium which follows, it is
explained that if the velocity is variable, then the
increment of the velocity "gained in the time AB
will not be the same with its Fluxion above defined,
. . . but if the Time AB be infinitely small, then
though the Velocity of x at the Instant B be
not the same, mathematically speaking, with the
Velocity at the Instant A ; yet the Difference being
infinitely small in Respect of the whole Velocity, it
may safely be neglected, where the finite Ratios of
Fluxions are only considered ; and so this Increment

1 The Method of Fluxions Applied to a select Number of Useful
Problems. ... By Nicholas Saunderson, Late Professor of Mathematics
in the University of Cambridge. London, MDCCLVI.

198 LIMITS AND FLUXIONS

and the Fluxion above defined may be taken for
one another, i.e. the Quantity x for so small a
Time, may be looked upon as flowing uniformly"
(p. 2). Later we read (p. 4) that if the times are
infinitely small, the quantity vx will be " infinitely
less " than vx or xv. Here the fluxions x, vx^ are
looked upon as infinitely small.

In the account of the life of Nicholas Saunderson,
printed in the first volume of his Elements of Algebra ,
Cambridge, 1740, p. xv, we read: "Our Professor
would not be induced by the Desires and Expecta-
tions of any, to engage in the war that was lately
waged among Mathematicians, with no small Degree
of Heat, concerning the Algorithm or Principles of
Fluxions. Yet he wanted not the greatest Respect
for the Memory of Sir Isaac Newton, and thought
the whole Doctrine entirely defensible by the strictest
Rules of geometry. He owned indeed that the
great Inventor, never expecting to have it canvassed
with so much trifling Subtility and Cavil, had not
thought it necessary to be guarded every where by
Expressions so cautious as he might have otherwise
used."

John Rowning , 1/56

1 74. A graduate of Magdalene College, Cambridge,
and a Fellow there, Rowning interested himself chiefly
in natural philosophy, but wrote also A Preliminary
Discourse x on fluxions, with the intention of writing

1 A Preliminary Discourse to an intended Treatise on the Fluxionary
Method. By John Rowning, M. A. London, 1756.

TEXT-BOOKS OF MIDDLE OF CENTURY 199

a full treatise. But the treatise in question never
appeared. After a popular exposition of the ideas
of fluxion and fluent, and of Leibniz's infinitely little
quantities and their summation, showing how these
methodsyield important results in natural philosophy,
he refers to Berkeley's attacks and the defence made
by Philalethes Cantabrigiensis, Walton, and Robins,
also Maclaurin, who ' ' declined entering the Combat,"
but endeavoured to treat the subject "in a Manner
less exceptionable." "But no Body, that I know
of," continues Rowning, "has explained it in so
easy and familiar a Way as I apprehend the Subject
capable of." Moreover, Jurin and Walton "carry
things ... no farther than Sir Isaac had done
before. They leave them, as to the Objections
made by the Analyst, exactly as they found them."
The difficulties do not lie in the idea of a first
fluxion — a velocity. "In this there is Nothing
either infinitely great or infinitely little : Nothing
obscure." As to higher fluxions, "these Things
indeed elude our Senses ; but they do not surpass
the Understanding" (p. 85). Berkeley's objection
to "infinitely small Quantities" is not fatal,
"because finite Measures might have been made
use of." His other objection, that "such Quanti-
ties are in some Cases retained and made use of for
a while, and afterwards, to use his own Expression,
like Scaffolds to a Building, are rejected as of no
Significancy, " may be met by the proof that those
quantities "are always such as ought by no means
to be retained." In further explanation of his

200 LIMITS AND FLUXIONS

position Rowning says (p. 88), "that the Velocity
of any Body is the same at any one Point, or at any
one Time, whether the Body moves with an uniform,
accelerated, or with a retarded Motion at that Point
or Time." This is elucidated by reference to geo-
metric figures, and amounts, in the main, to the
explanation given by Rowe in finding the fluxion
of xy. One objection to such explanations, which
had been raised by Berkeley, was that one could not
speak of the velocity a body had at a point of space.
That such a phraseology is admissible is tacitly
assumed by Rowning. What the latter emphasises
is that no use is made of the concept of the
" infinitely little." As to Berkeley's second objec-
tion, that the supposition which is made at the
beginning of the process is later displaced by its
contrary, as when the symbol o is at first made an
actual increment and later in the same process taken
as no increment, Rowning argues that terms involv-
ing factors oo, ooo, etc., "do arise in consequence
of the Acceleration wherewith the Power of x flows,
when x itself flows uniformly ; and consequently
that they arise from the second and higher Fluxions
of that Power ; and that, therefore, when the first
Fluxion of that power is only inquired after . . .
they are to be left out and rejected, as appertain-
ing to another Account." It can hardly be claimed
that Rowning made a contribution to the theory
of fluxions. However, he has a pleasant way of
expressing himself. His book was favourably re-
viewed in the Monthly Review (vol. xiv, p. 286).

TEXT-BOOKS OF MIDDLEOF CENTUR Y 201

Israel Lyons, 1758

175. Lyons was a mathematician and botanist.
His Treatise of Fluxions, London, 1758, is dedi-
cated to Robert Smith, Master of Trinity College,
Cambridge, "being the first Essay of a young and
unpractised Writer " which ' ' owed its first rude
Beginning to the early Encouragement" received
from the Master, as the author modestly states.
His treatment is geometric. He says: "I reject
no Quantities as infinitely smaller than the rest,
nor suppose different Orders of Infinitesimals and
infinitely great Quantities. But consider the Ratio
of the Fluxions as the same as that of the con-
temporaneous Increments, and take Part of the
Increment before and Part after the Fluent is
arrived at the Term, where we want the Fluxion,
since it is not the Increment after, or the Increment
before that we want, but at the very instant, which
can no otherwise be found but by considering Part
of the Increment before and Part after" (Preface).
Fluxions are defined as velocities. * ' The moments
of quantities are the indefinitely small parts, by the
addition or subtraction of which, in equal particles
of time, they are continually increased or diminished. "
The author proves the proposition: "The indefi-
nitely small spaces described in equal indefinitely
small times are as the velocities," since, "when
the time is diminished ad infinitum, the difference
of the velocities at the beginning and ending of
that time will vanish/' If two flowing quantities

202 LIMITS AND FLUXIONS

x and y are to each other in a given ratio, then in
xy = z it is argued that 2j/ = incr. of z + incr. of
x=3^x\ hence 2—2yx. Whenj/ = ^, this becomes

z=2xx; one has also, fluxion x+y =

From this is derived the fluxion of any rectangle

_ 2

xyt thus : The fluxion of x^ or x2-+2ry+.glP, is also
equal to 2xx+2yy + fluxion of 2xy. Hence fluxion
of 2xy — 2xy + 2xy.

" In the same manner as the quantities x, y, z, are
conceived to flow, and to have their fluxions, so
may the quantities x, j>, zt be supposed to be variable,
and therefore have their fluxions, which are thus
represented xy j/, z, and are called the second fluxions
of x,y, z" (p. n). "The fluent of any quantity as
x"'x is represented thus

William West, 1762

176. William West's Mathematics^- is a posthum-
( ous work ; the author died in 1760. Fluxions are

treated from the earlier Newtonian standpoint,
infinitely little quantities being used. Some novelty
is claimed for this text in the treatment of maxima
and minima.

James Wilson, 1761

177. In 1761 Wilson collected some of Benjamin
Robins's mathematical tracts in a two-volume book,

1 Mathematics. By the late Rev. Mr. Wm. West of Exeter. Revised
by John Rowe, London, 1762. There appeared a second, corrected,
edition in 1763.

TEX T-BOOKS OF MIDDLE OF CENTUR Y 203

entitled Mathematical Tracts of the late Benjamin
Robins. In an Appendix, Wilson inserts some
matters of historical interest regarding certain
manuscripts of Newton ; Wilson also defends Robins
against criticisms passed by a French writer, and
states his views of Maclaurin's indebtedness to
Robins.

Newton's Method of Fluxions (see our § 149) was
brought out in Paris in 1740 by George Louis Le
Clerk, Comte de Buffon, under the title, La methode
des Fluxions ', et des suites infinies. Buffon prepared
a historical Preface, in which he criticised severely
Berkeley and Robins for presuming to take excep-
tion to anything Newton had written on fluxions or
to modify Newton's mode of exposition. Buffon
praises Jurin, and then speaks of Robins thus
(pp. xxvii-xxix) :

". . . il commence par le censurer & par des-
aprouver sa maniere trop breve de presenter les
choses ; ensuite il donne des explications de sa
facon, & ne craint pas de substituer ses notions
incomplettes aux Demonstrations de ce grand
homme. II avoue que la Geometric de 1'Infini
est une science certaine, fondee sur des principes
d'une verite sure, mais enveloppee, & qui selon lui
rta jamais ete bien connue ; Newton n'a pas bien Id
les Anciens Geometres, son Lemme de la Methode
des Fluxions est obscur & mal exprime . . . :
malheursement les Mathematiciens ont ete plus
incredules que jamais, il n'y a pas eu moyen de leur
faire croire un seul mot de tout cela, de sorte que

204 LIMITS AND FLUXIONS

Philalethes comme defenseur de la verite, s'est
charge de lui signifier qu'on n'en croyoit rien,
qu'on entendoit fort bien Newton sans Robin, que
les pensees & les expressions de ce grand Philo-
sophes sont justes & tres-claires . . ., ce sont des
pieces d'une mauvaise critique. ..."

Buffon presents no argument against the views
expressed by Robins, but abuses him for presum-
ing to think independently. This doting attitude
toward Newton is justly attacked by James Wilson,
in his Appendix to the Mathematical Tracts of the
late Benjamin Robins , vol. ii, London, 1761, pp.
325-327. Wilson rightly says that if it was a
crime for Robins to make mention of the great
brevity with which Sir Isaac Newton wrote, Robins
was followed in it by Maclaurin and Saunderson.
"The truth is," says Wilson, "Sir Isaac Newton
at first made the same use of indivisibles, others
had done : in his Analysis per cequationes numero
terminomm infinitas, he expressly says, * Nee vereor
loqui de unitate in punctis, sive lineis infinite
parvis x ; ' and in his Lectiones Opticce he demon-
strated by indivisibles." Wilson contends further-
more that Buffon is wrong in claiming that the
mathematicians paid no regard to what Robins
had said, that in fact "the best writers soon
after trod in Mr. Robins's steps." In fairness
to Buffon it should be said, however, that he
printed his Preface in 1740, and that Maclaurin,
Saunderson, de Bougainville, and d'Alembert, whom

1 Comm. Epist, p. 85.

TEXT-BOOKS OF MIDDLE OF CENTUR ¥20$

Wilson mentions as following Robins, wrote at a
later date.

178. James Wilson claims1 that Maclaurin in
his Fluxions "conformed himself entirely to Mr.
Robins's sentiments in regard to Sir Isaac Newton's
doctrine," and "has even expressly followed his
plan in treating the subject." Jurin had contended
(says Wilson) "that Sir Isaac Newton's method, by
proving the varying quantities came up to their
limits, was more perfect than that of the ancients.
Whereas Sir Isaac Newton never claimed such
superiority ; . . . The coincidence contended for,
and thus highly praised by Philalethes, is the very
essence of indivisibles." Wilson rightly insists that
Buffon's criticisms of Robins are unfair. "When
he talks of the obscurity of Mr. Robins's ideas, the
insignificancy of his phrases, and the unintelligible-
ness of his style ; he gives the most certain proof,
that he had never carefully read his writings, . . .
for Mr. Robins is much admired here for the con-
trary excellencies, on whatever subjects he has
employed his pen."

179. Wilson represents Philalethes (Jurin) as
championing the use of the infinitely little and of
indivisibles. This is putting the case too strongly.
In his papers against Berkeley, Jurin uses quantities
infinitely little. But toward the end of his debate
with Robins he begins to disavow them. Never
did Jurin use indivisibles. Few eighteenth-century

1 Mathematical Tratts of the late Benjamin Robins, vol. ii, London,
1761, pp. 312, 315, 320.

206 LIMITS AND FLUXIONS

writers have brought out as distinctly and clearly
as has Jurin the difference between infinitesimals
as variables, and indivisibles ; Jurin disavowed all
quantity "fixed, determinate, invariable, indivisible,
less than any finite quantity whatsoever," but he
usually did admit somewhat hazily a quantity
"variable, divisible, that, by a constant diminu-
tion, is conceived to become less than any finite
quantity whatever, and at last to vanish into
nothing."

Remarks

1 80. None of the works mentioned in this chapter
are great works. Those of William Emerson and
Thomas Simpson were the best and the ones most
widely used. The first edition of Simpson is of
earlier date (1737).

CHAPTER VIII

ROBERT HEATH AND FRIENDS OF EMERSON IN
CONTROVERSY WITH JOHN TURNER AND
FRIENDS OF SIMPSON

1 8 1. The principals, Simpson and Emerson, do
not themselves appear in this controversy. During
the period of this debate, Robert Heath was editor
of The Ladies' Diary, which appeared once every
year as an almanac. We begin with one of his
articles.

Robert Heath, 1746

182. In an article, Of the Idea, and Nature of
Fluxions,^ Heath says :

' ' The Distinction betwixt the Increments and
Fluxions of Magnitudes, has been this ; that the
former approach in Ratio infinitely near the latter,
so that their Difference is unassignable. . . . What
we call the Fluxions, or Velocities of Magnitudes,
are only the Fluxions in Chief, or in Part, with
which they are born ; the Part neglected in the
Ratio exactly corresponding with what is rejected
in the finite Ratio of the infinitely small Increments,
which is therefore the same as the Ratio of our

1 The Ladies' Diary : or, the Woman's Almanack, for 1746.
207

208 LIMITS AND FLUXIONS

Fluxions.. And hence, whether we call those finite
Ratios, Fluxions, or Increments, their Idea, Nature,
and Original appear to be the very same thing.
For all Things are relative. ..."

He argues that while we consider a line or plane,
generating an area or solid, as of no thickness in the
mind, in our notation we represent them as of unit
thickness, ' * and consequently each Line or Plane
should be express'd by o x L, and o x P, to denote
them as they are in the Mind. But L x o to o,
and Px0 to o, are in the same Ratio with L to i,
and P to I, by equal Division by o ; and those again
in the same Ratio with L;r to £, and Pi: to x, by
equal Multiplication by x, for the Ratio of Fluxions.
But, this finite Notation of Line or Plane, which we
consider of no Breadth, or Thickness, and yet denote
by Unity, each, at the same Time, makes the Practice
and our Comprehension disagree. ... So that it will
be an Error to conclude that the Ratio of the
Fluxions of Quantities generated by the Motion of
Lines, or Planes, is arrived at this Way, without
the previous Consideration of an Increment ; for the
very Lines and Planes must be Increments, or Some-
things next to Nothings themselves, before they
were what we finitely express them by Notation,
or Quantities could never increase or be generated
thereby : For to carry a Line or Plane of no Breadth
or Thickness forward, is the same in Terms as to
carry Nothing forward. And therefore the Dis-
tinction between the Ratio of Increments, and that
of Fluxions, is only what the Conception of the

ROBERT HEATH v. JOHN TURNER 209

Thing differs from that of its Notation in
Practice. . . . Those who desire further Satisfaction
as to the Nature of Fluxions, of their noble Use
and transcendant Excellence, may consult Mr.
Emerson's Doctrine of the whole Art, which is ...
the best of any. . . . Those writers will find them-
selves mistaken, who pretended to derive the finite
Ratios of Motion, or Fluxions producing Magnitudes,
without the previous Consideration of Increments,
which include the very Notion of what a Fluxion is.
This some have attempted by multiplying Quanti-
ties into their Velocity, and some by other Means,
the Result of which originally depends on incre-
mental Principles, if they would consider the Matter
as far as it will go." The paper is brought to a
finish in the Ladies' Diary for 1747.

Main Articles in the Controversy

183. Over the pseudonym of " Cantabrigiensis "
there appeared in 1750 an unfriendly review of
Simpson's Doctrine and Application of Fluxions.^
The reviewer contended that the definition of a
fluxion as the "magnitude by which any flowing
quantity would be uniformly increased " (see our
§ 172) is very "odd"; for, "in quantities uni-
formly generated, the fluxion must be the fluent
itself, or else a part of it." Simpson's endeavour
to exclude ' * velocity " * * cannot be made intelligible
without introducing velocity into it." "Again, he

1 Monthly Review; or, New Literary Journal, vol. iv, London,
1750, pp. 129-131.

210 LIMITS AND FLUXIONS

mistakes the effect for the cause ; for the thing
generated must owe its existence to something, and
this can only be the velocity of its motion ; but it
can never be the cause of itself, as his definition
would erroneously suggest. " Moreover, it is strange
that Simpson "should still stick in the mud and
run himself into the old exploded method used by
foreigners ; and which is subject to all the cavils
that have ever been raised against that science."

184. This criticism originated a small tempest.
In a journal called Mathematical Exercises?- its
editor, John Turner, makes certain "Observations
on certain invidious Aspersions on Mr. Simpson's
Doctrine and Application of Fluxions, published in
the Monthly Review for December last, by Canta-
brigiensis." Mr. Simpson is there charged as having
' ' mistaken the Effect for the Cause " ; Mr. Simpson,
says Turner, "builds upon his own Definition;

1 Mathematical Exercises No. Ill (1751), p. 34. Six numbers of this
journal appeared in London in 1750-1752. No. V bears the date 1752 ;
No. VI has no date. Readers are invited "to send their Performances
(whether new Problems, Paradoxes, Solutions, etc.) Post paid, to
be left with Mr. James Morgan, at the Three-Cranes, in Thames-
street . . ." In this connection a statement made by Charles Hutton,
in his Memoirs of the Life and Writings of the Author [Thomas
Simpson], printed in Thomas Simpson's Select Exercises in the
Mathematics, new edition, London, 1792, p. xviii, is of interest:

"It has also been commonly supposed that he [Thomas Simpson]
was the real editor of, or had a principal share in, two other periodical
works of a miscellaneous mathematical nature ; viz. the Mathematician,
and Turner's Mathematical Exercises, two volumes, in 8vo, which
came out in periodical numbers, in the years 1750 and 1751, etc. The
latter of these seems especially to have been set on foot to afford a
proper place for exposing the errors and absurdities of Mr. Robert Heath,
the then conductor of the Ladies Diary and the Palladium ; and which
controversy between them ended in the disgrace of Mr. Heath, and
expulsion from his office of editor to the Ladies' Diary, and the substi-
tution of Mr. Simpson in his stead, in the year 1753."

ROBERT HEATH v. JOHN TURNER 211

which, he tells us, himself, is not exactly the same
as that of Sir Isaac Newton." Mr. Simpson is also
charged with plagiarism from Cotes's Estimatio
Errorum. John Turner says :

"Here his Remarks on the Author's Definition
of a Fluxion first demand our Consideration : Mr.
Simpson makes it to be, ' the Magnitude by which
a flowing Quantity would be uniformly increased in a
given Time.' This Definition the Critic represents
as a very old one ; and with regard thereto advances
the two following, extraordinary, Positions :

(< i. That, in Quantities uniformly generated, the
Fluxion must (according to the said Definition) be
the Fluent itself, or else a Part of it.

"2. And that, in other Quantities generated by
a variable Law, the Fluxion will not be a real, but
an imaginary Thing.

' ' To the first of these Objections I answer, that
the Fluxion is neither the Fluent itself nor a Part
of it : it is a Quantity of the same Kind with the
Fluent ; but the Fluent being the Quantity already
produced by the generating Point, Line or Surface,
supposed still in Motion, and the Fluxion what will
arise, hereafter, from the Continuation of that
Motion ; the latter can no more be denominated
a Part of the former than the ensuing Hour a Part
of the Time past.

4 'But his second Observation is a still more
glaring Instance of his Disingenuity, and Want of
Judgment. Does it follow, because a Body, really,
moves over a certain Distance, in a given Time,

212 LIMITS AND FLUXIONS

with an accelerated, or a retarded Velocity, that
there is no Distance over which it might pass in
the same Time, with its first Velocity uniformly
continued. The Space over which a Body would
uniformly move with such, or such, a proposed
Velocity, is no less real because no Part of it is
actually described with that Velocity " (pp. 36, 37).

185. Then follows an article reprinted from the
Daily Gazetteer for December 4, last [1750], in which
one who signs himself "Honestus" (said to have
been John Turner himself) charges that the compiler
of the Ladies' Diary (Robert Heath) is also the
compiler of the Palladium, and the best material
designed by contributors for the Diary are reserved
by him for the Palladium ; that the latter publica-
tion is owned by the compiler, while the former is
not. Robert Heath wrote a reply in the Daily
Gazetteer of December 6 ; four letters follow on
this subject.

1 86. John Turner's defence of Simpson led to the
publication of what Turner called a "scurrilous
Pamphlet." This pamphlet is without doubt the
Trutli Triumphant: or Fluxions for the Ladies^
London, 1752, or else those parts in that pamphlet
which appear over the pseudonyms "X Primus"

1 The fuller title of the pamphlet is thus: Truth Triumphant: or,
Fluxions for the Ladies. She-wing the Cause to be before the Effect , and
different from it; That Space is not Speed, nor Magnitude Motion.
With a Philosophic Vision, Most humbly dedicated to his Illustrious
High and Serene Excellence, the Sun. For the Information of the
Public^ by X, Y, and Z, who are not of the Family of xt y, s, but
near Relations of x1 y' and z' . . . . London. Printed for W. Owen.
M.DCC.LII.

ROBERT HEATH v. JOHN TURNER 213

and "Y Secundus." These documents evidently
emanated from the pen of Robert Heath, assisted
possibly by some other adherents of William
Emerson. At the risk, perhaps, of not observing
strict chronological sequence, we proceed to the
consideration of all parts of Truth Triumphant.
In the dedication "to the Sun," it is stated that
"the Family of the Wou'd-be's in this Island is
become very numerous, by uniformly continuing in
their Errors." Thus, both the title-page and the
dedication play on Simpson's definition of fluxions
and its alleged defects. In the Preface one reads :
"Fluxions, then, Ladies, that have so puzzled our
wise Mathematicians to define, are the respective
Degrees of Motion, at any Instant of Time, of any
two things or Bodies that continually flow, or move
on, over Space." Four pages are devoted to the
explanation of fluxions.

187. Then follow the two criticisms of John
Turner's defence of Simpson, signed "X Primus"
and "Y Secundus," to which we have alluded above.
In the former of these articles John Turner is
treated with contempt. "Who this John Turner
is, whether he is Mr. Simpson's Clerk, or his Pupil,
or some Dependant on him ; or whether he be
Mr. Simpson himself, is not very material to the
Reader ..." Turner is continually referred to
as "John." To Turner's reply to the first criticism
on Simpson's text, "X Primus" makes rejoinder:
"John says, the Fluent being the Quantity already
produced — Pray how was this Quantity produced,

214 LIMITS AND FLUXIONS

by some magic Art, without any Fluxion ? I believe
not. . . . For my Part, I know of no Body that
ever said, that the Parts of the Fluent that went
before were generated by the Fluxion that is to come
after, but every Part by its proper Fluxion. ..."
To Turner's reply to the second criticism, "X
Primus " makes rejoinder :

" If there be no Magnitude by which the flowing
Quantity is really increased, such a Magnitude is not
real, but an imaginary Thing only . . . But John
thinks, that every Thing that exists in his Imagina-
tion, really exists in Nature . . . Sir Isaac Newton
defines Fluxions by the Velocities of the Motions.
But Mr. Simpson declares against this, and likewise
tells us, that by taking Fluxions for mere Velocities,
the Imagination is confin'd, as it were, to a Point.
How his Imagination is confin'd I don't know ; but
Sir Isaac Newton chused to define it thus, as very
well, knowing, that this is the only solid Foundation
upon which it could be defended against all the
impertinent Cavils of ignorant or weak Pretenders."
The parting shot by * ' X Primus " is — your Great
Master will not " think you a fit Champion to
engage in his Cause for the future ; so, good Night,
John."

1 88. The reply made by " Y Secundus " is to the
effect that the defender of Simpson is "equally in
the Dark" with Simpson himself, "otherwise he
would not have gone about to defend so defenceless
a Cause, as to vindicate an Absurdity, by repre-
senting a Fluxion to be of the same Kind with the

ROBERT HEATH v. JOHN TURNER 215

Fluent, uniformly generated ; when the one is a
Quantity of actual Velocity, and the other a Quantity
of Space, described by that Velocity, which can be
only proportional to it."

189. After some poetry "To the Family of the
the Wou'd-be's," follow "Animadversions on Mr.
Simpson's Fluxions," "By 2 Tertius," who quotes
a criticism of Simpson from the pen of J. Landen.
Where Landen's review first appeared we do not
know. As quoted here, Landen objects to the
definition of fluxions ' ' as faulty, by the Author's
different Idea given of them to that by the
Inventor"; Landen disapproves of "denoting all
Quantities whatsoever by Lines, to bring them to
one Denomination, and those Lines, to be described
by Bodies in Motion." In criticism of fluxions in
general, Landen says that the finding, from the
velocities, the spaces passed over, and vice versa,
"may be managed by common Algebra, without
the least Obscurity. The Business had always
been better considered in that Light, without ever
making Use of the Term Fluxions, as if a new Kind
of Analysis, tho', in Fact, only the Doctrine of
Motion improved, and applied to Purposes before
unthought of."

190. The next article in Truth Triumphant is a
reprint of the first criticism of Simpson, contributed
in 1750 by Cantabrigiensis to the Monthly Review.
Eight more articles concerning motion, fluxions, and
mechanics bring the pamphlet to a close ; they
make no reference to Simpson, " Heliocentricus "

216 LIMITS AND FLUXIONS

explains higher fluxions in a way that cannot be
called illuminating.

191. Then " Amicus " speaks " of the Use of the
Algebraic Cypher, in finding the Fluxions of Alge-
braic Quantities," letting x increase or decrease, and
become x±o, where the "Increments or Decre-
ments are seen to be ±0," and "dividing by
algebraic 0," thereupon "algebraically considering
o of insensible Value, as before it was consider'd of
real sensible Value." Taking a reminiscent mental
attitude, " Amicus " says :

* * This algebraic Ratio of the Fluxions of Quantities,
to which the diminishing Value of the algebraic
Increments or Decrements, from their limited State
or Value, tend together, to their geometrical vanish-
ing (by supposing the variable Value less and less)
has been misconceived, as vanishing together with
the real geometrical Increments or Decrements they
are the Value of\ whence o has been denominated
a departed^ instead of an algebraic Quantity, by a
famous B— /, tho' it's Reality and Presence still
existed before his Eyes ; but if <?, the Cypher- Value,
or algebraic Quantity, call'd also Nothing •, be made
to signify Nothing, because it is so call'd, the Word
Nothing with as much Propriety, may be called no
Word, be allowed to have no Signification among
other Words, and be deem'd a mere Blank, as no
Subject capable of Consideration." Further on in
the pamphlet, the query whether there can be " real
Motion in no Time," for "any one Point of inter-
mediate Space gone over; especially since an infinite

ROBERT HEATH v. JOHN TURNER 217

Number of Points can never actually constitute real
Magnitude," and whether "Motion, or Fluxion,
can actually exist, and be known, but by the next
Increment of Space gone over, in some real and
next -Moment of Time"? These are fundamental
problems indeed. Zeno is not mentioned in the
pamphlet, but the query involves Zeno's subtle
paradox of the ' ' arrow. " Nor is the answer given
devoid of interest. " But Time, and Motion, flow-
ing over Space, . . . (since no Quantity can be
assigrid, or imagirid so small, but there will still be
smaller) the respective Degree of Velocity of Motion,
or Fluxion (i.e. instantaneous velocities) of that
Flowing at any Instant of Time, and Point of Space
gone over, will be everywhere assignable by the
immediate Increments, as Effects of those preceding
Velocities, as has been shewn. Whence it will
follow, that certain Degrees of Motion, Fluxion, or
Velocity, exist at every instant of Time taken, and
Point of Space respectively described ; contrary to
the differential Notion that Foreigners have of this
Matter." The weak spot here resides in the words
"immediate Increments"; do immediate increments
exist in view of the statement in the above paren-
thesis ? The lack of a satisfactory arithmetical con-
tinuum comes to view more fully in the antagonism
between geometric increases and algebraic increases
exhibited in the following passage taken from the
pamphlet :

"All the Values of the geometrical Increases
flow'd over, in finite Time, can never be algebraically

2i8 LIMITS AND FLUXIONS

express'd in infinite Time ; in which Sense the
algebraic Increases being again diminish'd, are said
never to converge to the Limits of their geometrical
Magnitudes in Motion, but will still have sensible
Value ; yet supposing the geometric Increases, and
their algebraic Values to flow and decrease alike,
to something determinate, then 0, and o, and it's
Powers into a Variable Quantity, and it's Powers,
will accurately express the Limits of variable Quan-
tities, or Beginnings of their Increases ; which
Limits, or Beginnings of Increases of Quantities,
are accurately, as the Fluxions of those Quantities
in general."

" Visionarius" closes with a philosophic vision in
which four candidates for honors appear before the
Goddess of Science. Rejected are the first three,
viz. the author of Mathematical Exercises (John
Turner), the one holding to the motto "A cypher
is no Algebraic Quantity " (Bishop Berkeley), and
a Grand Magnifier of Fluxions (Thomas Simpson) ;
crowned is the author of the incomparable treatises
of Fluxions and Trigonometry (William Emerson),
in whose interests Truth Triumphant appears to be
mainly written.

192. The probability is that the " scurrilous
pamphlet " to which John Turner made reply in
an issue of his Mathematical Exercises was only a
part of what is given in Truth Triumphant. The
latter is probably a later and enlarged publication.
In that reply Turner argues that " it must appear to
everyone that, what Mr, Simpson defines as Fluxions,

ROBERT HEATH v. JOHN TURNER 219

are exactly such finite Quantities, proportional to
the Velocities as Sir Isaac Newton here * speaks of ;
since it is well known that the Quantities produced,
or the Distances described, in any given Time, by
Motions uniformly continued, are, accurately, as
the Velocities of the said Motions."

193. In No. V (1752) of the Mathematical
Exercises appears another article in the contro-
versy, written by John Turner. There is little in
it requiring our attention. It is a reply to two
pamphlets, the Ladys Philosopher and a new
Palladium , both publications from the pen of
Robert Heath.

Ladies' Diary > 1751, 1752

194. The Ladies' Diary for 1751 has an article on
The Nature and Use of the Algebraic Cypher, or
Quantity o, "by Fluxioniensis " ; o / o is proved to
signify "any Value at Pleasure by considering
(a*— xn)-±(a — x) for n= I, 2, 3, 4, etc., when .*• = #. "
This " confutes the Notion of some Mathematicians"
that o / o expresses " a Ratio of Equality." Next
it is argued that o°=i. "Hence," says a second
anonymous critic, * ' all Cypher-Paradoxes, and
Mysteries of Ultimate Ratios, or Ratios of Least
Increments or Decrements of Quantities, vanish and
Day appears. ..."

" Waltoniensis, making a Distinction between
o signifying some Quantity, and o signifying no
Quantity, or absolute Nothing, says that when x

1 Principia, Bk. II, Lemma 2. See our §§ 16-19.

220 LIMITS AND FLUXIONS

converges to 0, in the ultimate State before it
vanishes, xx = I ; but says, when^r entirely vanishes,
or becomes absolutely of no Value, that then
x* = o° = o: But o being supposed no Quantity is
contradicted by Algebraic Computation, which is
general and retains o, in a Mathematical Sense, for
a Quantity in the Scale, as much as any other
Figure or Literal by which Quantity is denoted and
compared. . . . Waltoniensis farther observes that
Fluxions are the Limits to which the Ratios of the
Increments or Decrements of Quantities converge,
and are assignable from the Principles of Motion
only (uniform, accelerated, and retarded) ; and thinks
the Doctrine has nothing to do with infinitely small
Quantities, First and Last Ratios ; and that only
finite Quantities need be introduced — 'to avoid
Disputes, and the dark Mists spread over the Pro-
cess, different to the demonstrative Lights of the
Antients.' But Motion refers to the Spaces passed
over, by which it is comprehended, measured, and
compared : And tho' Mr. Simpson has pretended to
deduce the Ratios of Fluxions of Quantities without
the use of indefinitely small Quantities (see his New
Doctrine and Application of Fluxions} yet the Motion
of his Points along the Lines answers to them by
the indefinitely small Spaces described together,
and are to the same Effect as Quantities taken in-
definitely small ; which Sir Isaac Newton himself
introduced to illustrate the Quantity of relative
Motion by. Fluxions, as instantaneous Velocities,
are also as the Increases or Quantities of Space

ROBERT HEATH v. JOHN TURNER 221

passed over together by those instantaneous Veloci-
ties, uniformly continued ; but are not the Spaces or
Quantities themselves that would be described by
them according to Mr. Simpson's new Theory, and
Application. See Emerson's Doctrine and Applica-
tion of Fluxions. Price 6s. only. "

195. In the Ladies' Diary for 1752 the reader
is amused by satirical remarks on mathematicians.
There is also a continuation of the discussion " Of
the Cypher-Value and Office of the Algebraic
Quantity o." "Nihil Maximus says that '9999,
etc. ad inf. will never converge to I, nor yet
I / (10,000, etc. ad inf.) to o; because any Quantity
infinitely increased or diminished will be still greater
or less, and never numerically arrive at Infinity, and
o Value. . . . That a Distinction should be care-
fully made between what are called infinite and
indefinite great and small Quantities (the former of
which being impossible} ; for what is of indefinite
Value has Equality, tho' it may be sometimes un-
assignable ; while what is infinite is never determin-
able, and has never Equality. Hence the numerical
Value of o and i /(iooo, etc.), will be for ever
different ; one being a Quantity of no sensible
Value, but yet significant, and the other of indefi-
nite small Value. . . . Infinite Quantity, or in-
finite numerical Value, expressed by Authors, is
neither practicable nor comprehensible. . . . An
infinite Series can never precisely converge to a
finite or determinate Value ; because it for ever runs
on. The finite Value, taken for that of an infinite

222 LIMITS AND FLUXIONS

Series, is only the Value from whence that infinite
Series is or may be derived. Mr. Landen thinks
that the Value o is no algebraic Quantity ; but calls
it a mere Blank, or absolute Nothing . . . ; he
says, that its peculiar Office is only in arithmetic
Notation ; while we see it applied to other Use and
Office in Treatises of Algebra and Fluxions, as
also by himself, for an. algebraic Character or
Quantity o, its own Value."

This discussion of o is continued at great length.
Confusion arises from the double use of the symbol
and from the difficulties surrounding o as the limit
of variables or sequences. Reference is made twice
in this Ladies' Diary (1752) to the pamphlet Truth
Triumphant, or Fluxions for the Ladies, where the
nature and office of o are discussed, and exception
is taken to Landen's views on o. " Fluxioniensis "
says: "And therefore I should not stick to rank
this excellent Reasoner with the great Master of
Reason he mentions, the B — p of Cloyne, as he
clearly appears to be of the same class. "

Popular Impression of the Nature of Fluxions

196. A reviewer of Richard Jack's Euclid's Data
Restored^- quotes from Jack's preface what appears
to be the opinion of a non-specialist :

"Others, who claim the honour of extending
their principles, treat of what they call Fluxions,
calculus differential, infiniment petifs, extreme and

1 Alonthly Review, vol. xvi, London, 1757.

ROBERT HEATH v. JOHN TURNER 223

ultimate ratios, etc., and with so much obscurity,
that no distinct idea of the thing treated, is com-
municated to the mind of the Reader. From their
want of that precision and perspecuity which the
Ancients carefully observed in all their writings,
the mind becomes clouded with confusion, begins
to doubt, which terminates in a disbelief of their
principles ; for which reason they have been often
called upon to demonstrate them : but no demon-
stration has appeared."

To this the anonymous reviewer of Jack's book
replies :

"That the principles of Fluxions stand in need
of demonstration, especially since the publication of
Maclaurin's works, is certainly a mere pretence, made
only to cover the ignorance of the objector ..."

Remarks

197. In this chapter we have given views held by
writers representing the rank and file of mathe-
matical workers. In several passages the need of
an adequate theory of a linear continuum makes '
itself strongly felt.

Some curiosity attaches to the following contem-
poraneous opinion of Truth Triumphant : 1

4 'This is an odd assemblage of controversial
scraps, chiefly relating to some disputes concerning
Mr Emerson's treatise on fluctions, and Mr. Simp-
son's on the same subject. This most unimportant

1 Monthly Review, vol. v, London, 1751, p. 462.

224 LIMITS AND FLUXIONS

controversy was first occasioned by the few obser-
vations on Mr Simpson's book published in the
Monthly Review. . . . The author writes in a
manner that can do little honour to any party or
opinion. And why he chose to give this strange
and insignificant production so odd a title, is a
mystery that none but himself can clear up."

CHAPTER IX
ABORTIVE ATTEMPTS AT ARITHMETISATION

John Kirkby, 1748

198. In the preface to his Doctrine of Ultimators L
the author states that his doctrine " depends upon
scarce any Thing else but a due Application of the
Cypher o to the analogous Office in Universal
Arithmetic, which it is always known to occupy in
Common Arithmetic." He argues that "the super-
lative impropriety of the Word Fluxion, when
applied to this Purpose, will fully appear ; when we
come to consider, that it is put to express an Idea,
which arises from the Contemplation of Quantities
purely as Quantities : that is, in the same abstract
Manner, as they are the proper Subject of Algebra,
exclusive of every other Consideration ; and con-

1 The Doctrine of Ultimators. Containing a new Acquisition to
Mathematical Literature, naturally resulting from the Consideration of
an Equation, as reducible from its variable to its ultimate State : Or, a
Discovery of the true and genuine Foundation of what has hitherto mis-
takenly prevailed under the improper Names of Fluxions and the Differ-
ential Calculus. By means of which we now have that Apex of all
Mathematical Science entirely rescued from the blind and ungeometrical
Method of Deduction which it has hitherto laboured under ; and made to
depend upon Principles as strictly demonstrable ', as the most self-evident
Proposition in the first Elements of Geometry. By the Reverend Mr.
John Kirkby, Vicar of Waldershare in Kent. London, MDCCXLVIII.
Pp. 144.

225 I

226 LIMITS AND FLUXIONS

sequently have not the least Regard to Time or
Motion, which are necessarily implied in a Fluxion.
And the essential Property of a Fluxion is certainly
excluded, after the most singular Manner, in the
Idea of Quantity considered at its Ne plus ultra :
that is, in other Terms, when it is in a State,
where all Possibility of such imaginary Flux is
taken from it. So that the Term Fluxion, when
used to this Purpose, if it have any Meaning at all,
is as contrary to the true, as Darkness is to Light."
He takes an algebraic equation A^°±B<3:/±C^2zh
D#3db . . . ±Zan = o, assumes the coefficients Y
and Z of the two highest terms as fixed, and
declares (without proof) that the absolute term A
is a maximum when the n roots of the equation are
equal. When such an equality exists, the equation
is reduced "to its ultimate." When the roots are
equal he represents them by +c or — c. To reduce
the trinomial A±Ra±Zan = o to its ultimate, "we
must make B / Z = ^ncn~l in the n Power of c-±.a
= o. That is (because c=a)B / Z, = na"-1. There-
fore the Ultimate required is 'B±nZan~l = o, or
ftaO±nZan-laQ = o." To be observed* here is that
Kirkby connects, though only in an obscure way,
his ultimate with the coefficient of a in the second
term of the binomial expansion of (c±a)n. He then
pretends to prove "that the Ultimate of the Sum
of never so many Equations is the same with the
Sum of their respective Ultimates " ; hence, the
Ultimate of the above general equation is o±Ba°±

la° = o. He gives

ATTEMPTS AT ARITHMETISATION 227

the rule for finding the "ultimate" or "ultimator"
of a* ; this ultimator is na*~laQ ; he also gives the
rule for writing down * ' the subject of every Ulti-
mator" ; the subject of the ultimator nan~la° being
an + c. He applies these rules when the exponents
are fractional.

The " ultimator " of the product of two variables,
ae, is found thus. ' ' Put ae = bee, and ae = caa.
Whence a = be, and e — ca, and ae — \ bee + J caa.
The Ultimator of which last is ... &ee° + caa°, and
consequently is equal to the Ultimator of ae. But
be = at and ca = e. Therefore these substituted for
their Equals in that Ultimator give ae° + ea° for the
Ultimator of ae" (p. 43). It will, of course, be
noticed that special limitations are placed upon the
variables a and e, when the coefficients b and c are
tacitly assumed to be constants. Kirkby proceeds
to the derivation of the ultimators of fractions and
logarithms. He explains the necessity of retaining
in the Ultimator each variant (variable) under its
o Power. " Without this we cou'd have no Means
from the Nature of the thing itself, whereby to
distinguish an Ultimator from a Subject." The
functions of <7°, e° are more than simply to represent
unity ; just what they are is not very clear, although
to the author "it is evident then, as often as any
Subject consists of different Variants Ex gr. x> y, 2,
that the Expressions x°, j°, ZQ, in the Ultimator
have the same Difference in Power with the
same Variants under any other common Exponent
x", y, 2". . . . Therefore the Expression ^r°, j>°, s°t

228

LIMITS AND FLUXIONS

I conceive may be each fitly called the Peculiar Unit
of its respective Scale of Powers. Hence every
Ultiniator may be defined to be, The proper Reference
of each Subject in a given Equation to the Peculiar
Units of the Powers of all its Variants, in Order to
discover the Ratios of those Variants to one another in
their Ultimate State. Which 1 take to be the true
Definition of what has been hitherto most impro-
perly and unintelligibly called a
Fluxion by some, and a Differ-
ential by others " (page 49).

199. Kirkby's doctrine may
perhaps become plainer by the
study of one of his applications.
In any curve with the concave
side to TQ, the greater abscissa
VP (or v) has always the greater
" semi-ordinate " PM (or s\
"and each are the greatest
FlG n that they possibly can be to

the same Arch VM, or to the
same intercepted axis VR. Therefore the Sub-
normal PR (or r — v), and consequently the
Normal MR ( = c) are each the least that they
possibly can be to the same Arch VM, or the
same intercepted Axis VR (or r). Therefore, if
in the last Equation [r2 — <? = 2rv — v2— s2], c and
r be invariable, we have r2 — cz an Ultimum. Con-
sequently, the Ultimate of that Equation ... is
2rv° — 2vv° — 2ss° = o, or (dividing by 2) ss° = r — v x v°.
Whence v° : s° — s : r — v. That is in all Curves, as

ATTEMPTS AT ARITHMETISATION 229

the Ultimator of the Abscissa is to the Ultimator of
the Semiordinate ; so is the Semiordinate itself to
the Subnormal" (p. 51).

The author has occasion to use second and third
ultimators and to consider ultimators as variable or
invariable. He lets (p. 60) x° be the invariable of

the first ultimator x°, x the invariable of the second

ultimator x> etc., and warns the reader that his
dot does not mean a fluxion. In the more involved
applications to curves he lets an infinitely small arch
equal x°=x°. Our impression of the book is that
the author's intentions were good when he attempted
an arithmetisation. But there is a total lack of clear
and rigorous exposition.

200. The Ladies' Diary, London, 1750, p. 45,
contains a hostile criticism of the Doctrine of
Ultimators by an anonymous writer (probably the
editor, Robert Heath), in which the author of this
doctrine is said to declare that fluxions, as explained
by Newton, are " absurd and unintelligible," and to
place confidence in " the Authority of a certain Irish
B — p, a Mathematician as wise as himself. For
you must know that this pious B — p (the sagacious
Author of the Analyst, as he stiles him) out of his
religious Zeal against Mathematical Learning, had
been engaged in the same senseless Attempt with
himself, of degrading the noblest Science. . . .
Having thus, as he [the author of ''Ultimators"]
thinks, overturn'd the Doctrine of Fluxions . . .
he has given us instead of it ... a new Science of

230 LIMITS AND FLUXIONS

his own, whose Foundation, it seems, depends on
Cyphers, and Nought Powers full of conceited Ex-
pressions. . . , He expresses his Ultimators by
the Help of x°, jj/°, z°, etc., calls them peculiar Units,
and of different Values, all of which is absurd. . . .
I pass over ... his using x°t jj/°, ,5°, for the same
End as others use x, j, s. "

Jo Jin Petvin, 1750

20 1 . I n a Sketch of Universal A rithmetic, 1 brought
out as a posthumous booklet, we encounter acuriosity.
Its philosophy of mathematics and of fluxions in
particular is set off by the following quotations :

(Page 156) "I do not then consider it [mathe-
matical quantity] as generated or produced, but as
that which is. Time and Motion produce nothing
of the Kind, and have no Place here. Nor do I
consider it as continuous, nor as consisting of very
small or infinitely little Parts, but as consisting of
Parts in general. These Parts therefore I con-
sider as discrete : And by x, yy £, etc. , I under-
stand Multitude. The Ones or Monads, of which
x is many, I call x ; . . . Nor do I consider x, yy z,
etc. , barely as many ; but as a certain many. So
that x, j, £, etc. , are Wholes ; x, y, z, etc. , their
respective Parts. These Parts may be considered
again as Wholes, consisting of another Order of

1 Letters concerning Mind. To which is added, a Sketch of Uni-
versal Arithmetic ; comprehending the Differential Calculus, and the
Doctrine of fluxions. By the late Reverend Mr. John Petvin, A.M.,
Vicar of Ilsington in Devon, London, 1750.

ATTEMPTS AT ARITHMETISATION 231

Parts," designated as x^ etc. " Such Things as an
Instant, a Point, a Fluxion, she [arithmetic] has
nothing to do with. ... I have joined Fluxion with
Point and Instant, because Fluxion seems to be to
Motion, as an Instant is to Time ; which I suppose
to be as a Point is to a Line. Motion cannot be
conceived without Time and Space ; and when the
former runs into an Instant, and the latter into a
Point, then it is (as I understand it) that Motion
becomes Fluxion. ... In this Sense Fluxion is
no more a Part of Motion than a Point is a Part of
a Line." His " parts" are finite increments. The
part of xy is xy +yx + xy. " This Doctrine of Wholes
and Parts proceeds upwards from Parts to Wholes,
as well as downwards from Wholes to Parts uni-
versally" (p. 159). "The Ordinate therefore being
xm, when xmx expresses the Fluxion of it, the only
Meaning I have for x is, that it is the Proportion of
a Point to an Instant. And to my Apprehension,
a Point may as well be called a last Line, as this
called a Velocity." "I have lately deduced some
arithmetical Theorems from arithmetical Principles,
which other Mathematicians have drawn from
Fluxions of Fluxions, etc. , and these Theorems fell
in with my Design." Just how these deductions
were made is not explained by the author.

John Landen^ 1758

202. John Landen was a self-educated mathe-
matician of real mathematical power. Had he had
the benefits of University training he might have

232 LIMITS AND FLUXIONS

occupied a much higher rank as a mathematician.
Foreigners place him high among his English con-
temporaries. He wrote Mathematical Lucubrations,
1755, and Residual Analysis, 1764. We shall con-
sider only his Discourse concerning Residual Analysis ,l
1758. From it we quote as follows :

"Yet, notwithstanding the method of fluxions is
so greatly applauded, I am induced to think, it is
not the most natural method. . . . The operations
therein being chiefly performed with algebraic
quantities, it is, in fact, a branch of the algebraic
art, or an improvement thereof, made by the help
of some peculiar principles borrowed from the
doctrine of motion. . . . We may indeed very
naturally conceive a line to be generated by motion ;
but there are quantities . . . which we cannot
conceive .to be so generated. It is only in a
figurative sense, that an algebraic quantity can be
said to increase or decrease with some velocity.
Fluxions therefore are not immediately applicable
to algebraic quantities. ... It therefore, to me,
seems more proper, in the investigation of proposi-
tions by algebra, to proceed upon the anciently-
received principles of that art. . . . That the borrow-
ing principles from the doctrine of motion, with a
view to improve the analytic art, was done, not
only without any necessity, but even without any
peculiar advantage, will appear by showing, that
whatever can be done by the method of computa-

1 A Discourse Concerning Residual Analysis : A new Branch of the
Algebraic Art. By John Landen. London, 1758.

ATTEMPTS AT ARITHMETISATION 233

tion, which is founded on those borrowed prin-
ciples, may be done as well, by another method
founded entirely on the anciently -received principles
of algebra. . . . It is by means of the following
theorem [p. 5], viz.

v v
x^. — v^ ™-\ x x

Xn X

X— V

<">

(where m and n are any integers) that we are
enabled to perform all the principal operations in
our said Analysis."

His Residual Analysis is a method involving
vanishing fractions and therefore not free from con-
troversial questions. That the fluxion of x* is $x*
is explained according to the Residual Analysis by
the consideration that (j/3 — ^3)-rO/ ~~ x)=3?-\-xy+y2>
which is equal to $x2 when y—x. We proceed to
give an application in Landen's own words :

203. (Page 5) * ' Fluxionists, in determining the

limit of the ratio of the increments of x and xn ,
commonly have recourse to the binomial theorem
(which is much more difficult to investigate than the
limit they are seeking) : But how easily may that
limit be found, without the help of that theorem, by
the equation exhibited in page 5 ! Thus, the incre-
ment of x being denoted by x' > the increment of

xn is x+x'n —xn, and the ratio of those incre-
ments is

234 LIMITS AND FLUXIONS

x\ ~.i n

«-n

X- | Ji,

r+<

tr m

x+x1-

1+ *

2 x I3
1 f?tt^

x' ' x+x'

^r+y|

— X

2m ^ 'S*.

x+x'

^ x+x'

which, when y vanishes, is manifestly equal to

172 ^L — i

— xn , the limit of the said ratio."
n

The explanation of the method of drawing tan-
gents is too long for quotation, and we shall limit
ourselves to the following outline of it, as given
by Landen :

" I consider the curve as already described, with-
out any regard to its generation, and find the value
of a certain line (terminated by the curve and
its tangent), in algebraic terms involving (s) the
subtangent with other quantities ; which algebraic
expression I observe, from an obvious property of
the line it is found to denote, must have a certain
property with respect to being positive or negative
in certain cases. I therefore assume that expres-
sion equal to another which is known to have that
very property ; and from thence, by means of the
theorem mentioned in page 5, readily find the re-
quired value of s" (p. 10).

204. Landen's Discourse was attacked by an
anonymous writer in the Monthly Review for June
1759, who claims that the Residual Analysis "is no

ATTEMPTS AT ARITHMETISATION 235

other than Sir Isaac Newton's method of differences ;
and it is well known, that if the differences are
diminished so as to vanish, their vanishing ratio
becomes that of fluxions" (p. 560), "that his pre-
tended Residual Analysis renders the investigations
more tedious and obscure than any other. " Landen
wrote a reply in the July number, from which
we quote only the part relating to the word
* ' function. " Says Landen : ' * He objects to prime
number, function, etc., as terms never heard before. —
Alas ! how egregiously does he betray his ignor-
ance ! "

James Glenie, 1793

205. James Glenie graduated from the University
of St. Andrews, and became a military engineer.
He was a prominent Fellow of the Royal Society of
London. In his Antecedental Calculus^ 1793, he
begins with the statement, (( Having, in a Paper,
read before the Royal Society, the 6th of March,
1777, and published in the Philosophical Trans-
actions of that Year, promised to deliver, without
any consideration of Motion or Velocity, a Geo-
metrical Method of Reasoning applicable to every
purpose, to which the much celebrated Doctrine of
Fluxions of the illustrious Newton has been or can

1 The Antecedental Calculus, or a Geometrical Method of Reasoning,
without any Consideration of Motion or Velocity applicable to every
Purpose, to which Fluxions have been or can be applied. By James
Glenie, Esq., M.A. and F.R.S. London, 1793. According to G.
Vivanti (see M. Cantor's Vorlesungen uber Geschichte der Mathematik,
vol. iv, Leipzig, 1908, p. 667), James Glenie (1750-1817) was an
artillery officer in the war of the American Revolution, later professor
of mathematics in the military school of the East India Company.

236 LIMITS AND FLUXIONS

be, applied ; and having taken notice of the same
Method, in a small Performance, written in Latin,
and printed the i6th of July, 1776, I now proceed
to fulfil my promise with as much conciseness as
perspicuity and precision will admit of." In his
Antecedental Calculus, p. 10, he says of Newton :
* I am perfectly satisfied, that had this great Man,
discovered the possibility of investigating a general
Geometrical Method of reasoning, without introduc-
ing the ideas of Motion and Time, ... he would
have greatly preferred it, since Time and Motion
have no natural or inseparable connection with
pure Mathematics. The fluxionary and differential
Caculi are only branches of general arithmetical
proportion."

Glenie speaks (p. 3) of " the excess of the magni-
tude, which has to B a ratio having to the ratio of
A + N to B the ratio of R to Q (when R has to O
any given ratio whatever), above the magnitude,
which has to B a ratio having to the ratio of A to B
the same ratio of R to Q, is geometrically expressed
by" a complicated fraction whose denominator is
B(R-Q)/Q, and whose numerator is the result of ex-
panding by the binomial theorem (A + N)R/O- and
then subtracting AR/Q therefrom.

A similar expression is given for the case in which
A-N takes the place of A + N: "The excess of
the magnitude, which has to B a ratio, having to the
ratio of A to B the ratio of R to Q, above the magni-
tude, which has to B a ratio, having to the ratio of
A — N to B the ratio of R to Q, is geometrically

ATTEMPTS AT ARITHMETISATION 237

expressed by " a fraction whose denominator is
B(R-Q)/Q, and whose numerator is obtained by ex-
panding and simplifying AR/°- - (A - N)R/°-. "But
if A + N and A — N stand to B in relations nearer
to that of equality than by any given or assigned
magnitude of the same Kind, these general expres-
sions become R/Q . A<R-Q>/Q. N-^-B<R-Q>/Q. This I
call the antecedental of the magnitude which has
to B such a ratio as has to the ratio of A to B the
ratio of R to Q. Now if N the antecedental of A

be denoted by A or A ... [and] if Q = I and

2AA

3, 4, 5, etc., this expression gives -^— ,
. . respectively." For the "antecedent"

finds the "antecedental" ~ or 2 A

(putting M for A — B). Glenie shows that

at a point of a curve the antecedentals of the ab-
scissa, ordinate and curve, are as the sub-tangent,
the ordinate and the tangent, respectively.

Glenie's calculus involves extremely complicated
identities of ratios and examines the antecedents of
ratios having given consequents. The style of ex-
position is poor. In deriving the antecedentals,
Glenie quietly drops out all the terms in the
numerator that involve powers of N higher than
the first power. As this calculus plays no part in
the later history of fluxions, we shall give only one
more quotation ; it relates to the Binomial Theorem

238 LIMITS AND FLUXIONS

(not used by him in the development of his funda-
mental formulas). He says (p. 11) : "It may not
perhaps be improper to add, that, if to the ex-
pressions delivered above for the excess of the
magnitude, which has to B a ratio, having to the
ratio of A + N to B, the ratio of R to Q, above the
magnitude, which has to B a ratio, having to the
Ratio of A to B the same ratio of R to Q ; and for
the excess of the magnitude, which has to B a ratio,
having to the ratio of A to B the ratio of R to Q,
above the magnitude, which has to B a ratio, having
to the ratio of A- N to B the ratio of R to O, be
prefixed the magnitude, which has to B a ratio,
having to the ratio of A to B the ratio of R to Q,
we get a geometrical Binomial, of which, when it is
supposed to become numerical, the famous Binomial
Theorem of Sir Isaac Newton is only a particular
case."

Remarks

206. The classic treatment of fluxions in Great
Britain, during the eighteenth century, rests prim-
arily on geometrical and mechanical conceptions.
Attempts to found the calculus upon more purely
arithmetical and algebraical processes are described
in this chapter. All these attempts are either a com-
plete failure or so complicated as to be prohibitive.
Easily the ablest among these authors was John
Landen. De Morgan says of his Analysis1: "It
is the limit of D'Alembert supposed to be attained,

1 Penny Cyclopedia, Art. "Differential Calculus."

ATTEMPTS AT ARITHMETISAT1ON 239

instead of being a terminus which can be attained as
near as we please. A little difference of algebraical
suppositions makes a fallacious difference of form :
and though the residual analysis draws less upon
the disputable part of algebra than the method of
Lagrange, the sole reason of this is that the former
does not go so far into the subject as the latter."

In the same article De Morgan speaks of Kirkby's
Ultimators thus :

" A something between Landen and D'Alembert,
as to principle, published in 1748, was called the
1 Doctrine of Ultimators, containing a new Acquisi-
tion, etc., or a Discovery of the true and genuine
Foundation of what has hitherto mistakenly pre-
vailed under the improper names of Fluxions and
the Differential Calculus.'"

CHAPTER X
LATER BOOKS AND ARTICLES ON FLUXIONS

Encyclopedia Britannica, 1771, 1779, 17 97

207. The article * * Fluxions " in the first edition
of the Encyclopedia Britannica, Edinburgh, 1771,
gives this definition: "The fluxion of any magni-
tude at any point is the increment that it would
receive in any given time, supposing it to increase
uniformly from that point ; and as the measure will
be the same, whatever the time be, we are at liberty
to suppose it less than any assigned time." The
fluxion of a rectangle is the increment, with the small
rectangle at the corner omitted ; the latter c< is owing
to the additional velocity wherewith the parallelo-
gram flows during that time and therefore is no
part of the measure of the fluxion." "The incre-
ment a quantity receives by flowing for any given
time, contains measures of all the different orders of
fluxions ; for if it increases uniformly, the whole in-
crement is the first fluxion ; and it has no second
fluxion. If it increases with a motion uniformly
accelerated, the part of the increment occasioned by

the first motion measures the first fluxion, and the

240

LATER BOOKS AND ARTICLES 241

part occasioned by the acceleration measures the
second fluxion. ..."

The same article is reprinted in the second edition
(1779) and the third edition (1797).

Robert Thorp, 1777

208. Thorp made a translation of part of New-
ton's Principia.^

In the " advertisement " we read : " The doctrine
of prime and ultimate ratios ... is established, so
as to remove the various objections which have been
raised against it, since it was first published. To
the relations of finite quantities alone the reasoning
on this subject is confined." The translation of
quantitates quam minima, evanescentes, ultima, in-
finite magnce, and the like, has not been literal, yet
they are " explained in that sense under which the
author cautions his readers to understand them.
This is the more necessary, as the terms infinite,
infinitesimal, least possible, and the like, have been
grossly misapplied and abused."

209. In the Commentary to Lemma I in Sect. I
of Bk. I in the Principia, Thorp says : ' ' The prime
and ultimate ratios of magnitudes . . . are investi-
gated by observing their finite increments or decre-
ments, and thence finding the limits of the ratios
of those variable magnitudes ; not the ratios to
which the magnitudes ever actually arrive (for

1 Mathematical Principles of Natural Philosophy. By Sir Isaac
Newton, Knight. Translated into English, and illustrated with a
Commentary, by Robert Thorp, M.A., vol. i, London, 1777.

242 LIMITS AND FLUXIONS

they are never, strictly speaking, either prime or
ultimate in fact), but those limits to which the
ratios of magnitudes perpetually approach ; which
they can never reach, nor pass beyond ; but to
which they appear nearer than by any assignable
difference." . . . "We now proceed to explain this
Lemma more particularly than perhaps might seem
necessary, had it not been much controverted, mis-
represented, and misunderstood." As one of the
conditions of the proposition, Thorp states, is "that
quantities and the ratios of quantities must con-
tinually tend to equality. The one must never
become equal to, nor pass beyond the other : their
difference must never either vanish to nothing, or
become negative." In this restriction Thorp goes
even further than had Robins. The following
passage from Thorp's commentary is thoroughly in
the spirit of Robins : ' '. . . That we may not be led,
from the expression ultimately equal, to suppose,
that there is an ultimate state, in which they are
actually equal, we are cautioned in the scholium at
the end of this Section [of Principia^ Bk. I, Sect, i]
in these words, The ultimate ratios , in which quantities
vanish , are not in reality the ratios of ultimate
quantities ; but the limits to which the ratios of
quantities continually decreasing always approach ;
which they never can pass beyond, nor arrive at, unless
the quantities are continually and indefinitely dimin-
ished. According to Thorp, the inscribed or cir-
cumscribed polygon can never arrive at the curve.
He quotes from Saunderson's Fluxions. By the

LATER BOOKS AND ARTICLES 243

doctrine of indivisibles there "has been introduced
into mathematical reasoning all that absurd jargon
concerning quantities infinitely great, and infinitely
little, which has been so much objected to by mathe-
maticians. And, though it has often been elegantly
applied by some able geometers to the demonstra-
tion of many noble theorems ; yet in the hands of
less accurate reasoners, it has often led to false
conclusions" (p. 71).

F. Holliday , 1777

210. In a somewhat lengthy preface to his Intro-
duction to Fluxions^ the author tells that, when in
1745 he was in London, in company with W. Jones
and De Moivre, they expressed great approbation
of Emerson's Fluxions, with regard to the method
of treatment, but thought his book too high for
beginners. The author tries to be more diffuse in
the laying down of first principles. He derives the
fundamental results in two ways : first, by the aid
of nascent or evanescent quantities, as suggested by
Newton's Principia; second, " without using any
infinitely small quantities, or vanescent Parallelo-
grams, which perhaps will be more acceptable to
many of my Readers. " Holliday explains at great
length the Scholium (see our §§ 10-15) on prime
and ultimate ratios, and gives a short account of
the invention of fluxions as given in the review of

1 An Introduction to Fluxions, Designed for the Use, and Adapted to
the Capacities of Beginners. By the Reverend F. Holliday, Vicar of
West Markham and Bothamsall, Noll's. London, 1777.

244 LIMITS AND FLUXIONS

Collins's Commercium Epistolicum in the Philosophical
Transactions, 1717. Though following Newton
closely, variations were bound to arise. Thus,
Holliday says (p. 73), " Fluxions are not magnitudes
but the velocities with which magnitudes, varying by
a continual motion, increase or decrease." It cannot
be claimed that Holliday made any contribution to
the philosophy of fluxions, nor even that he profited
as much as he might by the refinements in the logic
which had been made by English writers since the
time of Newton.

Charles Hutton, 1796, 1798

211. In his Mathematical Dictionary, London,
1796, Charles Hutton makes reference to the
advantage of Simpson's definition of a fluxion as a
magnitude uniformly generated in a finite time, the
imagination being now no longer confined to a single
point and to the velocity at that point ; moreover,
' ' higher orders of Fluxions are rendered much more
easy and intelligible."

212. From the part on fluxions in Hutton's Course
of Mathematics 1 we take the following :

4 ' The rate or proportion according to which
any flowing quantity increases, at any position or
instant, is the Fluxion of the said quantity, at
that position or instant : and it is proportional to
the magnitude by which the flowing quantity would
be uniformly increased, in a given time with the

1 A Course of 'Mathematics. By Charles Hutton. London, 4th ed.,
1803-1804, vol. ii, p. 279. [First ed., 1798-1801.]

LATER BOOKS AND ARTICLES 245

generating celerity uniformly continued during that
time."

" . . . If the motion of increase be accelerated,
the increment so generated, in a given finite time,
will exceed the fluxion : . . . But if the time be
indefinitely small, so that the motion be considered
as uniform for that instant ; then these nascent
increments will always be proportional, or equal, to
the fluxions, and may be substituted instead of them
in any calculation."

The fluxion of xy is derived in two ways : the first
by the method of considering the rectangle composed
of two parts, as previously expounded by Rowe.

The second method finds algebraically the incre-
ment xy' ' +yx' '+^y ', "of which the last term xfy' is
nothing, or indefinitely small, in respect of the other
two terms, because x' and yr are indefinitely small
in respect of x and y. . . . Hence, by substitut-
ing x and y for x' and yf , to which they are propor-
tional, there arises xy+yx for the true value of the
fluxion of xy."

S. Vince, 1795, 1805

213. Vince's Principles of Fluxions appeared in
1795 as the second volume of the Principles of
Mathematics and Natural Philosophy in Four
Volumes -,1 which were brought out under the
general editorship of James Wood. A second

1 The Principles of Mathematics and Natural Philosophy in Four
Volumes. Vol. //, The Principles of Fluxions : Designed for the Use
of Students in the University. By the Rev. S. Vince, A.M., F.R.S.,
Cambridge, 1795.

246 LIMITS AND FLUXIONS

edition of Vince was printed in 1805. From this
second edition we quote :

P. i : ' ' The velocities with which flowing quan-
tities increase or decrease at any point of time, are
called \htftttxions of those quantities at that instant.

"As the velocities are in proportion to the
increments or decrements uniformly generated in
a given time, such increments or decrements will
represent the fluxions."1

Vince also quotes Newton on the generation of
quantities by motion : " Sir I. Newton, in the
Introduction to his Quadrature of Curves, observes
that ' these geneses really take place in the nature
of things, and are daily seen in the motion of bodies.
And after this manner, the ancients, by drawing
moveable right lines along immoveable right lines,
taught the geneses of rectangles. ' '

Vince gives no formal definition of a limit ; but
his philosophy of this subject is disclosed by the
two following quotations (pp. 4 and 5): "By
keeping the ratio of the vanishing quantities thus
expressed by finite quantities, it removes the
obscurity which may arise when we consider the
quantities themselves ; this is agreeable to the
reasoning of Sir I. Newton in his Principia, Lib. I,
Sect, i, Lem. 7, 8, 9."

" It has been said, that when the increments are

1 "This is agreeable to Sir I. Newton's ideas on the subject. He
says : ' I sought a method of determining quantities from the velocities
of the motions or increments with which they are generated ; and call-
ing these velocities of the motions or increments, flitxions, and the
generated quantities fluents, I fell by degrees upon the method of
fluxions.' — Introd. to Quad. Curves"

LATER BOOKS AND ARTICLES 247

actually vanished, it is absurd to talk of any ratio
between them. It is true ; but we speak not here
of any ratio then existing between the quantities,
but of that ratio to which they have approached as
their limit ; and that ratio still remains. Thus, let
the increments of two quantities be denoted by
ax* +mx and bx* + nx ; then the limit of their ratio,
when ;tr = o, is m:n\ for in every state of these
quantities, axz + mx : bx* + nx : : ax + m : bx+n : :
(when x = d) m : n. As the quantities therefore
approach to nothing, the ratio approaches to that
of m : n as it's limit. We must therefore be careful
to distinguish between the ratio of two evanescent
quantities, and the limit of their ratio ; the former*
ratio never arriving at the latter, as the quantities
vanish at the instant that such a circumstance is
about to take place."

By aid of the binomial theorem, Vince finds the
fluxion of xny when the fluxion of x is given ; he then
finds the fluxion of xy by considering (x+y)2=x2 +
2xy+yz, by which the fluxion of 2xy can be found
in terms of the fluxions (ar+jj/)2, x* and j2.

Agnesi — Colson — Hellins, 1 80 1

214. The Analytical Institutions^ is the first cal-
culus that was written by a woman. The authoress

1 Analytical Institutions, in four books : Originally written in
Italian, by Donna Maria Gat ana Agnesi, Professor of the Mathe-
maticks and Philosophy in the University of Bologna. Translated
into English by the late Rev. John Colson, M.A., F.R.S., and Lucasian
Professor of the Mathematicks in the University of Cambridge. Now
first printed, from the Translators Manuscript, under the inspection of
the Rev. John Hellins, B,D,, F.R.S. Vojs. i and ii. London, 1801,

248 LIMITS AND FLUXIONS

is the noted Maria Gaetana Agnesi, of the University
of Bologna.

The Italian original was first published at Milan
in 1748. The two volumes of the translation were
printed at the expense of Baron Maseres. In an
introduction, Hellins points out that Colson hoped
to interest the ladies of England in the study of
fluxions by his translation of the work of the great
Italian lady, " And, in order to render it more easy
and useful to the Ladies of this country, ... he
[Colson] had designed and begun a popular account
of this work, under the title of The Plan of the
Lady's System of Analyticks ; explaining, article by
article, what was contained in it. But this he did
not live long enough to finish."

215. Colson dealt with Agnesi's work somewhat
as Stone had dealt with that of De L'Hospital,
inasmuch as both translators substituted the nota-
tion of Newton in place of that of Leibniz. The
word fluxions (" flussioni") occurs in the original
Italian of Agnesi's masterly work. How Colson's
conscience may have troubled him, when a fluxion
stood out in his translation as something " infinitely
little," may be judged when we consider that in
1736 he brought out an English translation, with
an extensive comment, of Newton's Method of
Fluxions. With Newton a fluxion always meant
a velocity.

We quote a few passages from Colson's Agnesi
(vol. ii, pp. i, 2):

"The Analysis of infinitely small Quantities,

LATER BOOKS AND ARTICLES 249

which is otherwise called the Differential Calculus,
or the Method of Fluxions, is that which is con-
versant about the differences of variable quantities,
of whatever order those differences may be."

" Any infinitely little portion of a variable quantity
is called it's Difference or Fluxion ; when it is so
small, as that it has to the variable itself a less pro-
portion than any that can be assigned ; and by
which the same variable being either increased or
diminished, it may still be conceived the same as
at first. "

On p. 3 we read that certain lines in a figure
" will be quantities less than any that can be given,
and therefore will be inassignable, or differentials, or
infinitesimals, or, finally, fluxions. Thus, by the
common Geometry alone, we are assured that not
only these infinitely little quantities, but infinite
others of inferior orders, really enter the composi-
tion of geometrical extension."

"These propositions," says a reviewer1 of the
translation, " may appear exceptionable, in point of
language, to the rigorists in geometry ; but they are
nevertheless founded on good principles, and furnish
rules for the comparison of evanescent quantities,
which will prove safe guides in investigation. The
demonstrations appear to us to be perfectly sound
(if the word infinite be taken in its true sense, as
denoting merely the absence of any limit), with the
exception, perhaps, of the first theorem, which
(as is not a little curious to remark) is liable to the

1 Edinburgh Review, vol. iii, 1805, p, 405.

250 LIMITS AND FLUXIONS

same objection that has been made of .the lemma
of Newton's Principia. In both instances, also, the
error is rather apparent than real. " The first theorem
in question states that the two intersecting perpendi-
culars to a curve drawn at the ends of " an infinitely
little portion of it of the first order," "may be
assumed as equal to each other. " We wonder what
Robins and Maclaurin would have thought, had they
been alive in 1801 and 1805, and read these defini-
tions and comments ! What horrible visions would
these ghosts of departed quantities have brought to
Bishop Berkeley, had he been alive ! But the nine-
teenth century was destined to bring back to British
soil still greater accentuations of infinitesimals.

T. Newton, 1805

216. The Rev. T. Newton says in the preface of
his Illustrations of Sir Isaac Newton's Method : *

' * Every Mathematician now considers the whole
doctrine of Prime and Ultimate Ratios in no other
light, than as a Doctrine of Limits." Young
readers of Sir Isaac Newton's Principia encounter
difficulties because commentators have made "use
of the terms of Indivisibles, in their explana-
tions ; . . . Newton expressly says, that by the
ultimate ratios of quantities he means the ratios of
their limits.2 And when he wants to infer the

1 An Illustration of Sir Isaac Newtoris Method of Reasoning. By
Prime and Ultimate Ratios. By the Rev. T. Newton, Rector of Tewin,
Herts ; late Fellow of Jesus College, Cambridge. Leeds, 1805.

2 See our §§ 12, 15.

LATER BOOKS AND ARTICLES 251

equality of inequality of those limits from some
relation of the variable quantities, which are never
supposed absolutely to reach their limits, it cer-
tainly requires something more than a definition to
shew this. ... It is not my intention to detain
the reader, with answering the objections of the
Analyst and his followers, because it has been
already done by others in a satisfactory manner.
. . . Notwithstanding the assertions of some
modern writers, the method of ultimate ratios is
extremely perspicuous, strictly logical, and more
concise than any other of modern invention ; . . .
it neither involves the strange notion, that a
straight line may be a part of a curve, and a plane
superficies a part of a concave or convex one ; nor
the unintelligible idea of adding and subtracting
indivisibles, or inconceivably small magnitudes.
Whatever magnitudes are compared, according to
this method, they are always supposed to be
finite."

T. Newton begins with the following two defini-
tions (p. i) :

"If a variable quantity, either increasing or de-
creasing, approaches to a fixed quantity, the differ-
ence between them being continually diminished, so
as at length to become less than any assignable
quantity ; the fixed quantity is called the Limit of
the variable quantity."

" If the ratio 'of two variable quantities continu-
ally approaches to a fixed ratio, so as to come
nearer to it than by any assignable difference ; the

252 LIMITS AND FLUXIONS

fixed ratio is called the Limiting Ratio of the
variable quantities. "

William Dealtry, 1810, 1816

217. In the preface of Dealtry's Principles of
Fluxions^ (1816) we read:

"The method of Fluxions rests upon a principle
purely analytical ; namely, the theory of limiting
ratios ; and the subject may therefore be considered
as one of pure mathematics, without any regard to
ideas of time and velocity. But the usual manner
of treating it is to employ considerations resulting
from the theory of motion. This was the plan of
Sir Isaac Newton in first delivering the principles of
the method ; and it is adopted in the following
Work, from the belief, that it is well adapted for
illustration."

Dealtry defines a ' ' fluxion of a quantity at any
point of time" as "its increment or decrement,
taken proportional to the velocity with which the
quantity flows at that time." . . .

" When a quantity increases with a velocity which
continually varies, the quantity, which measures
the fluxion, is a limit between the preceding and
succeeding increments, and is ultimately equal to
either of them. " He explains that * ' the word
ultimately is intended to denote that particular
instant, when the time is diminished sine limite"

1 The Principles of Fluxions : Designed for the Use of Students in the
Universities. By William Dealtry, B.D., F.R.S., late Fellow of
Trinity College, Cambridge. 2nd ed., Cambridge, 1816,

LATER BOOKS AND ARTICLES 253

and quotes Newton's Scholium, Sect. I, in the Prin-
cipia. He points out, also, that if x increases uni-
formly, X* increases with accelerated velocity, and
the part of the increment x'2 is the effect of the
acceleration, and therefore, by his definition of
fluxion, to be "omitted in taking the fluxions"
(p. 8).

New Editions ', 1801-1809

218. William Davis, who was a bookseller in
London and editor of the Companion to the Gentle-
man's Diary, appears also as the editor of new
editions of three different texts on fluxions. In
1 80 1 he saw through the press the second edition
of Maclaurin's Treatise of Fluxions ; in 1 805 the
third edition of Thomas Simpson's Doctrine and
Application of Fluxions. In 1809 appeared the fourth
edition of John Rowe's Doctrine of Fluxions •, revised
"by the late William Davis."

Remarks

219. Among some of the authors of this period
there is less concern than among writers of former
years about the attainment of the rigour of the
ancients. Perhaps the effects of the revival of the
ideals of Euclid and Archimedes which followed the
publication of the Analyst were gradually subsiding.
It would not be fair to this age to judge its mathe-
matical status altogether by the authors which we
have selected. There was a movement under way

254 LIMITS AND FLUXIONS

at this time which is reflected in the literature that
will be under consideration in the next chapter.

Both before the time of Berkeley's Analyst and
after the time of Maclaurin's Fluxions there appeared
in Great Britain texts which superposed British
symbols and phraseology upon the older Continental
concepts. The result was a system, destitute of
scientific interest. Newton's notation was poor and
Leibniz's philosophy of the calculus was poor. That
result represents the temporary survival of the least
fit of both systems. The more recent international
course of events has been in a diametrically opposite
direction, namely, not to superpose Newtonian
symbols and phraseology upon Leibnizian concepts,
but, on the contrary, to superpose the Leibnizian
notation and phraseology upon the limit-concept, as
developed by Newton, Jurin, Robins, Maclaurin,
D'Alembert, and later writers.

CHAPTER XI

CRITICISMS OF FLUXIONS BY BRITISH WRITERS
UNDER THE INFLUENCE OF D'ALEMBERT,
LAGRANGE, AND LACROIX

Review of Lagrange's " Fonctions analytiques" 1799

220. Important is a review x of Lagrange's Theorie
des fonctions analytiques^ which, as is well known, is
an attempt to deduce the principles of the calculus,
diverted of all reference to infinitely small or evan-
escent quantities, limits or fluxions, and reduced to
the algebra of finite quantities. The reviewer gives
a general criticism of the methods of fluxions and the
differential calculus. He discusses the principle of
motion : " It will not be denied that this principle is
introduced purely for the purpose of illustration, . . .
on the ground of convenience. . . . The mathe-
matical principle, on which the doctrine of fluxions
depends, is a definition . . . and fluxions were
defined to be velocities. . . . Now velocity is
nothing real, but is only the relation between the
space described and the time of describing it ;— of
which relation we have a clear idea when the motion
is uniform." The reviewer continues : (( In variable

1 Monthly Review ', London, vol. xxviii, 1799, Appendix.
255

256 LIMITS AND FLUXIONS

motion, however, we inquire what velocity is ; and
here it is defined to be the relation between the
space which would be described were the motion
continued uniform from any point, and the time.
Still difficulties remained ; this definition might con-
vey to the mind a general idea of the nature of
velocity, but was of no mathematical use, since the
space which would be described could not be immedi-
ately ascertained and determined. Another step
was therefore to be made, and which was made by
establishing this proportion ; if V be the velocity,
S the space, which would be described, and T the

time, S' the . space really described, and T' the

S
corresponding time; then V = — = ultimate ratio of

, when S' and T' are indefinitely diminished."

Again he says :

" On the ground of perspicuity and evidence, the
understanding is not much assisted by being directed
to consider all quantity as generated by motion ;
. . . when such quantities as weight, density, force,
resistance, etc., become the object of inquiry . . .
then the true end of the figurative mode of speech,
illustration, is lost. . . . That which happened to
Aristotle has happened to Newton ; his followers
have bowed so implicitly to his authority, that they
have not exercised their reason. The method of
fluxions had never so acute, so learned, and so
judicious a defender as Maclaurin : — yet who-
ever consults it ... finds the author speaking of

CRITICISMS BY BRITISH WRITERS 257

'causes and effects,' of 'the springs and principles
of things/ and proposing to deduce the 'relation
of quantities by comparing the powers which are
conceived to generate them ' ; — will be convinced
that this could only happen from so able a mathe-
matician having failed to seize the right principles."
"If English mathematicians first adopted Newton's
method from veneration to him, . . . they have
since persevered in it (we may almost say) against
conviction. " The reviewer claims that the criticisms
of D'Alembert, Torelli, and Landen have shown that
the use of motion is unnecessary and unreal. We
have given citations from Landen in an earlier
chapter (see our §§ 202, 203). D'Alembert is
quoted as saying fifty years previous :

" Introduire ici le mouvement, c'est y introduire
une idee etrangere, et qui n'est point necessaire a la
demonstration : d'ailleurs on n'a pas d'ide"e bien nette
de ce que c'est que la vitesse d'un corps a chaque
instant, lorsque cette vitesse est variable. La Vitesse
n'est rien de reel ; . . . c'est le rapport de 1'espace
au terns, lorsque la vitesse est uniforme ; . . .
Mais lorsque le mouvement est variable, ce n'est
plus le rapport de 1'espace au terns, c'est le rapport
de la differentielle de 1'espace a celle du terns ;
rapport dont on ne peut donner d'idee nette, que
par celle des limites. Ainsi, il faut necessairement
en revenir a cette derniere idee, pour donner une
idee nette

1 Art. "Fluxion" in Encyclopedic, ou Dictionnaire raisonnt des
sciences^ etc., t. 6, Paris, 1756.

258 LIMITS AND FLUXIONS

221. The reviewer states that foreign mathe-
maticians have written treatises in which motion is
entirely excluded, "and in some of these treatises,
the principles of the doctrine in question have been
laid down with a considerable degree of evidence
and exactness." The Residual Analysis of Landen
rests on "a process purely algebraical: but the
want of simplicity ... is a very great objection to
it." The reviewer is of the opinion that Euler and
D'Alembert give " the most clear and precise notions
of the principles on which the differential calculus
is established." He refers to Euler's Institutiones
calculi differentiates, 1755. D'Alembert, says the re-
viewer, " observes that the method is really founded
on that of prime and ultimate ratios, or of limits,
which latter method is only an algebraical transla-
tion of the former ; that, in fact, there are no such
things as infinitely small quantities ; and that, when
such quantities are mentioned, it is by the adoption
of a concise mode of speech for the purpose of
simplifying and abridging the reasoning ; — that the
true object of consideration is the limit of the ratio
of the finite differences of quantities."

The reviewer continues : "The explanations given
by Euler and D'Alembert, beyond all doubt, deserve
much consideration, yet their method of consider-
ing the doctrine of fluxions is not completely satis-
factory, but is objectionable on two grounds : first,
that we have no clear and precise notion of the
ratio of quantities, when those quantities are in
their vanishing state, or cease to be quantities ;

CRITICISMS BY BRITISH WRITERS 259

secondly, the connection and natural order of the
sciences are interrupted, if we give a distinct and
independent origin to that which in fact, is a branch
of analysis derived from the same common stock,
whence all the other branches are deduced.'* Then
follows a sympathetic account of the foundations
for the calculus laid by Lagrange in his Theorie des
f auctions analytiques, 1798. In passing, the reviewer
remarks that " Emerson, Stone, Simpson, Waring,
etc. , have published treatises on fluxions ; in none
of which, however, are the principles clearly laid
down."

Review of a Memoir of Stockier, 1799

222. In the same journal1 there is a review of
a memoir on fluxions written by the Portuguese
mathematician, Garcao Stockier, who modifies the
explanation of fundamentals by the introduction of
a "hypothetical fluxion" (a uniform velocity that
generates a quantity equal to the real increment
generated during the actually variable motion), which
is always contained between the proper fluxions at
the first and second instant under consideration.
By diminishing the interval of time, the hypothetical
fluxion approaches the true fluxion more nearly than
by any assignable quantity. Here also, the real
object of consideration is the limit. The reviewer
argues that the fundamental principles are not
new, and that the objections to Newton's fluxions
apply equally to those of Stockier. In a reply to

1 Monthly Review > vol. xxviii, London, 1799, p. 571.

26o LIMITS AND ^FLUXIONS

the Monthly Review, Stockier denies the reviewer's
allegation that he [Stockier] supposed quantity to
be generated by motion. "The idea of motion,
and the idea of velocity, are too particular to be
admitted into a general theory of fluent quantities." x

Review of Lacroix's (( Calcul differentiel" 1800

223. A review of S. F. La Croix's Traite
du calcul differentiel2 served as the occasion of
further comments and criticisms of fundamental
concepts :

" Who would direct his ridicule against the refine-
ments, subtleties, and trifling of the schoolmen, if he
read what has been written by some men who were
presumed to be the greatest masters of reason, and
whose employment and peculiar privilege consisted
in deducing truth by the justest inferences from
the most evident principles ? The history of the
differential calculus, indeed, shows that even mathe-
maticians sometimes bend to authority and a name,
are influenced by other motives than a love of truth,
and occasionally use (like other men) false meta-
physics and false logic. No one can doubt this, who
reads the controversial writings to which the inven-
tion of fluxions gave rise : he will there find most
exquisite reasonings concerning quantities which
survived their grave, and, when they ceased to
exist, did not cease to operate ; concerning an in-
finite derivation of velocities, — and a progeny of

1 Monthly Review, vol. xxxii, p. 497.

2 Monthly Review, vol. xxxi, London, iSoo, p. 493.

CRITICISMS BY BRITISH WRITERS 261

infinitesimals smaller than the ' moonshine's wat'ry
beams,' and more numerous than

* Autumnal leaves that strow the brooks.
In Vallombrosa.' (Milton, Par. Lost, i, 302.)

' c The contemporaries and partizans of Newton
were men infinitely inferior to him in genius, but
they had zeal, and were resolved to defend his
opinions and judgments. Hence they undertook the
vindication of fluxions, according to the principles
and method of its author ; although it may be fairly
inferred, from the different explanations given of
that doctrine by Newton in different parts of his
works, that Newton himself was not perfectly satis-
fied of the stability of the grounds on which he
had established it."

The reviewer quotes (p. 497) from Lacroix's
preface :

"These notions [velocities, motions], although
rigorous, are foreign to geometry, and their applica-
tion is difficult. . . . Properly speaking, fluxions
were to him [Newton] only a means of giving a
sensible existence to the quantities on which he
operated. The advantage of the method of fluxions
over the differential calculus in , point of meta-
physics, consists in this ; that, fluxions being finite
quantities, their moments are only infinitely small
quantities of the first order, and their fluxions are
finite ; by these means, the consideration of in-
finitely small quantities of superior orders is avoided.
... I can only mention a method which Landen
gave in 1758, to avoid consideration of infinity of

262 LIMITS AND FLUXIONS

motions, or of fluxions, since it rests on a very
elegant algebraic theorem which cannot be given in
a work of this nature. The freedom with which
Landen divests himself of national prejudice stamps
a remarkable character on his work ; he is perhaps
the only English mathematician, who has acknow-
ledged the inconvenience of the method of
fluxions." . . .

"We can always descend from the function to
the differential coefficient or from the primitive
function to the derived function : but, generally
speaking, the reverse step is attended with the
greatest difficulty."

' ' The rivals of Newton thought and invented
for themselves ; had they been influenced by his
authority, and devoted their talents to the perfec-
tion of synthesis, science must have been con-
siderably retarded. To the improvement of the
algebraical analysis, is to be attributed the amazing
advances of physical astronomy. " *

Review of Carnofs "Reflexions" 1801

224. In the Monthly Review'2' (London) for 1801
there is a short and unimportant account of Lazare
N. M. Carnot's new book, Reflexions sur la meta-
physique du calcul infinitesimal ', 1/97. Carnot
explains the correctness of results obtained by the
infinitesimal calculus of Leibniz on the theory of
compensation of errors — a theory which had been

1 Monthly Review, vol. xxxii, p. 491.

2 Monthly Review, vol. xxxiv, 1801, p. 463.

CRITICISMS BY BRITISH WRITERS 263

advanced much earlier by Berkeley in his Analyst.
Mr Philip E. B. Jourdain has found clear indica-
tions of this theory in Maclaurin's Fluxions and in
Lagrange's Theorie des fonctions analytiques. The
method of limits is explained by Carnot in the
manner of D'Alembert. "Of fluxions, indeed,"
says the reviewer, "as founded on the strange basis
of velocity, there is no account. "

Robert Woodhouse^ 1803

225. In 1803, Robert Woodhouse published his
Principles of A nalytical Calculation. x Woodhouse had
graduated B. A. at Caius College, Cambridge, in
1795, as senior wrangler. He then held a scholar-
ship and a fellowship at Caius College, devoting
himself to mathematics. He has the distinction of
being the first to strongly encourage the study in
England of the mathematical analysis which had
been created on the Continent by Swiss and French
mathematicians. In his Principles of Analytical
Calculation he discussed the methods of infinitesi-
mals and limits, and Lagrange's theory of function,
pointing out the merits and defects of each. " By
thus exposing the unsoundness of some of the
Continental methods, he rendered his general support
of the system far more weighty than if he had
appeared to embrace it as a blind partisan."2

226. The ideas set forth in this book are, on the

1 The Principles of Analytical Calculation^ by Robert Woodhouse,
A.M., F.R.S. Cambridge, 1803.

2 Art. " Woodhouse, Robert," in Sidney Lee's Dictionary of National
Biography.

264 LIMITS AND FLUXIONS

whole, in such close agreement with those advanced
in the preceding reviews, that the query naturally

ises, whether Woodhouse is not the author of
those reviews. We have reached no final decision
on this point.

In the preface Woodhouse passes in review the
different methods of establishing the foundations of
the calculus. He criticises the use of motion in the
proof of the binomial and other related theorems.
(( It required no great sagacity to perceive, that a
principle of motion, introduced to regulate processes
purely algebraical, was a foreign principle." If the
binomial theorem and related theorems for the
development of a function be established by algebra,
independently of motion, then ' * from the second
term of this expansion, the fluxion or differential of
a quantity may be immediately deduced, and in a
particular application, it appears to represent the
velocity of a body in a motion. The fluxionists
pursue a method totally the reverse ; they lay down
a principle of motion as the basis of their calculus,
thence deduce some of the first processes, and
establish the binomial theorem, by which it is said,
the extraction of roots may be effected. . . . The
project of extracting the square and cube roots of
algebraical quantities by a principle of motion, is
surely revolting to the common sense."

" Of his own method, Newton left no satisfactory
explanation : those who attempted to explain it,
according to what they thought the notions of its
author, and ... by reasoning which fairly may be

CRITICISMS BY BRITISH WRITERS 265

called tedious and prolix. Of the commentators on
the method of fluxions, Maclaurin is to be esteemed
most acute and judicious, but his Introduction
exhibits rather the exertions of a great genius
struggling with difficulties, than a clear and distinct
account of the subject he was discussing." To
remove this prolixity, it was proposed, conformably
to the notions of Newton, to call in the doctrine of
prime and ultimate ratios or of limits. Euler and
D'Alembert, on the other hand, rejected motion, but
retained limiting ratios, failing, however, in supply-
ing a satisfactory explanation therefor. Wood-
house is the earliest English mathematician who
speaks in respectful and appreciative terms of
the services to mathematics rendered by Bishop
Berkeley. In fact, Woodhouse admits as valid
some of Berkeley's objections which had been
declared invalid. The methods of treating the
calculus ' ' all are equally liable to the objection of
Berkeley, concerning the fallacia suppositions, or
the shifting of the hypothesis." Thus, in fluxions
and the method of limits, x is increased by /, and,
in the case of x"lt the increment of the function,

divided by *, is Mx»-1 + m(m~~I)*m-*t + ,etc.; then,

putting, 2 = 0, there results mxm~l. But since the
expansion of (x+i)m was effected "on the express
supposition, that i is some quantity, if you take
z = O, the hypothesis is, as Berkeley says, shifted,
and there is a manifest sophism in the process "
(p. xii).

266 LIMITS AND FLUXIONS

227. As another objection to limits, or prime
and ultimate ratios, Woodhouse declares that "the
method is not perspicuous, inasmuch as it considers
quantities in the state, in which they cease to be
quantities."

Moreover, ' ' the definition of a limit, is neither
simple nor concise" (p. xvii). ' ' The name of Berkeley
has occurred more than once in the preceding pages :
and I cannot quit this part of my subject without
commending the Analyst and the subsequent pieces,
as forming the most satisfactory controversial dis-
cussion in pure science, that ever yet appeared : into
what perfection of perspicuity and of logical pre-
cision, the doctrine of fluxions may be advanced, is
no subject of consideration : But, view the doctrine
as Berkeley found it, and its defects in metaphysics
and logic are clearly made out. If, for the purpose
of habituating the mind to just reasoning ... I
were to recommend a book, it should be the Analyst."
"The most diffuse and celebrated antagonists of
Berkeley, are Maclaurin and Robins, men of great
knowledge and sagacity : but the prolixity of their
reasonings confirms the notion, that the method they
defend is an incommodious one."

" Landen, I believe, first considered and proposed
to treat the fluxionary calculus merely as a branch of
Algebra : After him, M. Lagrange, a name ever to
be celebrated, in the Berlin Acts for 1772, laid
down its analytical principles ; and subsequently in
his Theorie des fonctions analytiques, 1796, he has
resumed the subject : in this treatise, the author

CRITICISMS BY BRITISH WRITERS 267

expressly proposes, to lay down the principles of the
differential calculus, independently of all considera-
tion of infinitely small, or vanishing quantities, of
limits, or of fluxions " (p. xviii). While Wood-
house considers Lagrange's discussion as very valu-
able, he does not find it free from logical faults.

William Hales, 1804

228. As a protest against the new movement and
a vindication of Newton from the attacks upon
fluxions in the Monthly Review, William Hales pre-
pared a book, the Analysis Fluxionum, which was
published in Maseres' Scriptores Logarithmici, vol. v,
London, 1804. Hales endeavours to show that the
doctrine of prime and ultimate ratios is really the
same as the doctrine of the limits of the ratios.
Hales's fundamental definitions are :

" Rationes ultimae sunt limites, ad quos quanti-
tatum sine fine decrescentium rationes, i, semper
appropinquant ; et, 2, quas propius assequi possunt
quam pro data quavis differentia ; 3, nunquam vero
transgredi ; 4, nee prius attingere, quam quantitates
ipsae diminuuntur in infinitum."

"Momentum est fluentis augmentum aut decre-
mentum momentaneum; id est, tempore quam minimo
genitum. Estque fluxioni proportionale. "

After Hales's work had gone to press, he became
acquainted with Benjamin Robins's Discoursed 1735,
and published in appendices1 numerous extracts from

1 Maseres, Scriptores Logarithmici, vol. v, pp. 848, 854, 856.

268 LIMITS AND FLUXIONS

it. Says Hales : " It is far superior indeed to the
subsequent explanations of professed commentators ;
and it is a high gratification to myself to find, that
the mode of explanation, which I adopted of the
Doctrine of Limits, is precisely the same as Robins's ;
long before I had seen his admirable treatise, which
did not fall into my hands until lately, a considerable
time after the publication of the Analysis Fluxionum. "
Maseres calls the Discourse of Robins "the ablest
tract that has ever been published on the subject."
Hales's text and the appendices to it contain con-
siderable historical material, consisting mainly of
references to and quotations from earlier writings.
In view of the testimony of Laplace, Legendre, and
Lacroix on the superiority of the method of fluxions,
1 'how was it possible," asks Hales, that the eyes of
the Monthly Reviewers "could still be so holden
... as still to assert, that Newton himself was not
perfectly satisfied of the stability of the ground on
which he had established his Method of Fluxions ! "
Hales's motive in opposing Continental ideas was
probably partly theological. D'Alembert, con-
sidered by him a hostile critic of Newton, is called
"a philosophizing infidel," one " of the original con-
spirators against Christianity," "at once the glory
and disgrace of the French Academy of Sciences,"
whose last words were "a terrific contrast to
the death of the Christian Philosopher," Colin
Maclaurin.1

The publication of Hales's Fluxions in large

1 Maseres, Scriptores Logarithmici^ vol. v, pp. 176-182.

CRITICISMS BY BRITISH WRITERS 269

quarto form and in the Latin language, the in-
clusion in the Appendix of matters foreign to the
subject of the book, together with the attempt
to maintain a system of notation and mode of
exposition that was beginning to be considered
provincial, caused the book to "fall still-born
from the press."

Encyclopedia Britanntca, 1810

229. In the fourth edition of the Encyclopedia
Britannica, Edinburgh, 1810, the article "Fluxions"
is wholly rewritten, and is much more extensive than
the article in former editions. There is a lengthy
historical introduction, and emphasis is placed upon
work done on the Continent. It observes "that
there is no work in the English language that ex-
hibits a complete view of the theory of fluxions,
with all the improvements that have been made
upon it to the present time." Mention is made
then of "several excellent works in the French
language," mentioning Cousin, Bossut, La Croix,
L'Huilier.

Letting u be " any function " of ^r, the limit of
the ratio (u — u) f h is defined as "a quantity to
which the ratio may approach nearer than by any
assignable difference, but to which it cannot be con-
sidered as becoming absolutely equal." The article
asserts that the method of fluxions "rests upon a
principle purely analytical, namely the theory of
limiting ratios ; and this being the case, the subject
may be treated as a branch of pure mathematics,

2;o LIMITS AND FLUXIONS

without having occasion to introduce any ideas
foreign to geometry. Sir Isaac Newton, however,
in first delivering the principles of the method,
thought proper to employ considerations drawn from
the theory of motion. But he appears to have done
this chiefly for the purpose of illustration, for he
immediately has recourse to the theory of limiting
ratios, and it has been the opinion of several mathe-
maticians of great eminence (such as Lagrange,
Cousin, La Croix, etc., abroad, and Landen in this
country) that the consideration of motion was intro-
duced into the method of fluxions at first without
necessity, and that succeeding writers on the subject
ought to have established the theory upon principles
purely mathematical, independent of the ideas of
time and velocity, both of which seem foreign to
investigations relating to abstract quantity." "By
the fluxions then of two variable quantities having
any assigned relation to each other, we are in the
following treatise always to be understood to mean
any indefinite quantities which have to each other the
limiting ratio of their simultaneous increments (we
. . . mean the ratio of the numerical values of the
increments, which may always be compared with
each other, whether the variable quantities be of
the same kind, as both lines, or both surfaces, etc.,
or of different kinds, as the one a line, and the other
a surface). The Newtonian notation is used in the
article exclusively."

CRITICISMS BY BRITISH WRITERS 271

Lacroix's ' ' Elementary Treatise ', " 1 8 1 6

230. The translation of Lacroix's Elementary
Treatise on the Differential and Integral Calculus 1
in 1816 marks an important period of transition.

From the " Advertisement" we quote :

This work of Lacroix " may be considered as an
abridgement of his great work on the Differential
and Integral Calculus, although in the demonstra-
tion of the first principles, he has substituted the
method of limits of D'Alembert, in the place of the
more correct and natural method of Lagrange,
which was adopted in the former. The first part
of this Treatise, which is devoted to the exposition
of the principles of the Differential Calculus, was
translated by Mr. Babbage. The translation of the
second part, which treats of the Integral Calculus,
was executed by Mr. G. Peacock, of Trinity College,
and by Mr. Herschel, of St. John's College, in nearly
equal proportions."

On p. 2 the process of differentiation of u = axz
is explained, so that 2ax "is the limit" of the ratio
(u — u) I /i, or it is "the value towards which
this ratio tends in proportion as the quantity h
diminishes, and to which it may approach as near
as we choose to make it."

Thus Lacroix's definition, like D'Alembert's, does
not prohibit the limit to be reached. In Note A,
added by the translators, we read :

1 An Elementary Treatise on the Differential and Integral Calculus.
By S. F. Lacroix. Translated from me French. Cambridge, 1816.

272 LIMITS AND FLUXIONS

" A limit, according to the notions of the
ancients, is some fixed quantity, to which another
of variable magnitude can never become equal,
though in the course of its variation it may approach
nearer to it than any difference that can be
assigned." Thus, the method of limits is here
ascribed by the translators to the ancients, which
is an act of reading into the ancient expositions a
theory not actually there. The ancient " Method
of Exhaustions " is merely a prelude to the theory
of limits. Peacock gives in Note A a history of
the theory of limits, in which researches on the
Continent are dwelled upon and the contribution
made by Newton is explained, but no reference is
made to Jurin, Robins, and Maclaurin. In Note B
Peacock states that the method used by Lacroix in
this treatise "was first given by D'Alembert, in
the Encyclopedic " article * * Differential. " Evidently
Peacock was not altogether friendly toward this
method, for in Note B he proceeds ' ' directly to
show in what manner this calculus may be estab-
lished upon principles which are entirely indepen-
dent of infinitesimals or limits," and then informs
the reader "that we are indebted for the principal
part of the contents of this note, to the Calcul des
Eonctions of Lagrange and the large treatise by our
author, on the Differential and Integral Calculus."
Peacock proceeds to give an account of Lagrange's
calculus of functions and of the method of fluxions.
Attention is called to "the difficulty of denoting
the operations of finding the different orders of

CRITICISMS BY BRITISH WRITERS 273

fluxions " according to the Newtonian notation,
"when for u we put the function itself, which it
represents. "

23 1. The attitude of some British mathematicians
of the early part of the nineteenth century toward
the discussions of the fundamental concepts of the
calculus carried on during the eighteenth century
is exhibited in the following passage from John
Leslie's Dissertation on the progress of mathematical
and physical science : x

' 'The notion of flowing quantities, . . . appears
on the whole, very clear and satisfactory ; nor
should the metaphysical objection of introducing
ideas of motion into Geometry have much weight.
Maclaurin was induced, however, by such cavelling,
to devote half a volume to an able but superfluous
discussion of this question. As a refinement on the
ancient process of Exhaustions, the noted method
of Prime and Ultimate Ratios . . . deserves the
highest praise for accuracy of conception. It has
been justly commended by D'Alembert, who ex-
pounded it copiously, and adapted it as the basis of
the Higher Calculus. The same doctrine was like-
wise elucidated by our acute countryman Robins ;
. . . Landen, one of those men so frequent in
England whose talents surmount their narrow
education, produced in 1758, a new form of the
Fluxionary Calculus, under the title of Residual
Analysis, which, though framed with little elegance,

1 Dissertation Fourth, in the Encyclopedia Britannica, 7th ed.,
vol. i, 1842, pp. 600, 60 1.

274 LIMITS AND FLUXIONS

may be deemed, on the whole, an improvement on the
method of ultimate ratios."

Remarks

232. The first part of the nineteenth century
marks a turning-point in the study and teaching of
mathematics in Great Britain. Attention has been
directed to the efforts of Woodhouse to introduce
the higher analysis of the Bernoullis, Euler, Clairaut,
and Lagrange. His efforts were strongly and ably
seconded by three other young men at Cambridge,
John Frederick William Herschel, Charles Babbage,
and George Peacock, who used to breakfast together
on Sunday mornings, and in 1812 founded the
"Analytical Society at Cambridge," for the promo-
tion, as Babbage humorously expressed it, of " the
principles of pure D-ism in opposition to the Zto/-age
of the University." The translation into English of
Lacroix's Elementary Treatise and the publication,
in 1820, of Examples with their solutions, brought
the more perfect notation of Leibniz and the re-
fined analytical methods to the attention of young
students of mathematics in England.1

1 Before the nineteenth century, the use in England of the Leibnizian
notation dz andyj/dk is exceedingly rare. In our § 54 we saw that
about the beginning of the eighteenth century these symbols were used
by John Craig in articles published in the London Philosophical
Transactions. When criticising Euler, Benjamin Robins once used
the Leibnizian notation ; see our § 142. Mr. Philip E. B. Jourdain has
brought to my attention the fact that the sign of integration f occurs
also in a book, entitled, Second Volume of the Instructions given in
the Drawing School established by the Dublin Society. , . . Under the
Direction of Joseph Fenn, heretofore Professor of Philosophy in the
University of Nantes. Diiblin, MDCCLXXII. De Morgan refers to
this work in a letter to Hamilton. See Graves' Life of Sir William
Rowan Hamilton , vol. iii, p. 488. See also our Addenda, p. 289.

CRITICISMS BY BRITISH WRITERS 275

As usually happens in reformations, so here, some
meritorious features were discarded along with what
was antiquated. William Hales, in 1804, referred
to the much neglected Discourse of Benjamin Robins \
(!735)> with its full and complete disavowal of!
infinitesimals and clear-cut, though narrow, con-
ception of a limit. By a curious turn in the
process of events, Robins was quite forgotten in
England, and D'Alembert's definition was recom-
mended and widely used in England, Now Robins
and D'Alembert had the same conception of a limit ;
both held to the view that variables cannot reach
their limits. However, there was one difference
between the two men : Robins embodied this restric-
tion in his definition of a limit ; D'Alembert omitted
it from his definition, but referred to it in his
explanatory remarks. D'Alembert says : x

' ' On dit qu'une grandeur est la limite d'une autre
grandeur, quand la seconde peut approcher de la
premiere plus pres que d'une grandeur donnee, 'si
petite qu'on la puisse supposer, sans pourtant que la
grandeur qui approche, puisse jamais surpasser la
grandeur dont elle approche ; ensorte que la differ-
ence d'une pareille quantite a la limite est absolu-
ment inassignable." Further on in the same article
we read : "A proprement parler, la limite ne co-
incide jamais, ou ne devient jamais egale a la
quantite dont elle est la limite ; mais celle-ci s'en

1 Art. ' ' Limite " in the Encyclopedic, ou dictionnaire raissonnt des
Sciences des arts el des metiers, publit par M. Diderot, et M. D* Alembert.
Paris, 1754. See also the later edition of Geneva, 1772.

276 LIMITS AND FLUXIONS

approche toujours de plus en plus, & peut en differer
aussi peu qu'on voudra. . . . On dit que la somme
d'une progression geometrique decroissante dont le
premier terme est a & le second b, est (a — b) / (ad) ;
cette valeur n'est poit proprement la somme de la
progression, c'est la limite de cette somme, c'est-a-
dire la quantite dont elle peut approcher si pres
qu'on voudra, sans jamais y arriver exactement."

233. That even the best expositions of limits and
the calculus that the Continent had to offer at that
time were recognised in England to be imperfect, is
shown by a passage in a letter which William Rowan
Hamilton wrote in 1828 to his friend John T.
Graves : 1

"I have always been greatly dissatisfied with
the phrases, if not the reasonings, of even very
eminent analysts, on a variety of subjects. . . . An
algebraist who should thus clear away the meta-
physical stumbling-blocks that beset the entrance
of analysis, without sacrificing those concise and
powerful methods which constitute its essence and
its value, would perform a useful work and deserve
well of Science."

1 Life, of Sir William Rowan Hamilton^ by Robert P. Graves, vol. i,
1882, p. 304.

DEFECTS 279

the theory of

ind by such

acroix, it

ike of

CHAPTER XII

MERITS AND DEFECTS OF THE EIGHTEENTH-
CENTURY BRITISH FLUXIONAL CONCEPTIONS

Merits

234. There are, perhaps, no intuitional concep-
tions available in the study of the calculus which
are clearer and sharper than motion and velocity.
There is, therefore, a certain advantage in approach-
ing the first study of the differential calculus or of
fluxions by the consideration of motion and velocity.
Even in modern teaching of the elements to
beginners, we cannot afford to ignore this advantage
offered by the eighteenth-century British mode of
treating the calculus.

A second point of merit lies in the abandonment
of the use of infinitely little quantities. Not all
English authors of the eighteenth century broke
away from infinitesimals, but those who did were
among the leaders : Robins, Maclaurin, Simpson,
Vince, and a few others. The existence of infini-
tesimals (defined as infinitely small constants) was
looked upon by philosophers and by many mathe-
maticians as doubtful. Their subjective existence
was hardly more probable than their objective exist-
ence. These mystic creations occupied a hypo-

277

2?6 LIM rS AND FLUXIONS

che "^ight zone between finite quantity and
• .itity. Their abandonment added to the
,, ness and logical rigour of mathematics. From
r .e standpoint of rigour, the British treatment of
the calculus was far in advance of the Continental.
It is certainly remarkable that in Great Britain
there was achieved in the eighteenth century, in the
geometrical treatment of fluxions, that which was
not achieved in the algebraical treatment until the
nineteenth century ; it was not until after the time
of Weierstrass that infinitesimals were cast aside
by many mathematical writers on the Continent.

235. There is a perversity in historic events
exhibited in the fact that after infinitesimals had
been largely expelled in the eighteenth century
from Great Britain as undesirable, unreal, and
mischief-making, they should in the nineteenth
century be permitted to return again and to flourish
for a time as never before. About 1816 the
Leibnizian notation of the calculus and the vast
treasures of mathematical analysis due to the
Bernoullis, Euler, D'Alembert, Clairaut, Lagrange,
Laplace, Legendre, and others, which were all ex-
pressed in that notation, found their way into
England. This influx led to enrichment and advance-
ment of mathematics in England, but also to a
recrudescence — this return of the infinitely small.
How thoroughly the infinitesimal invaded certain
parts of British territory is seen in Price's large
work on the Infinitesimal Calculus, a work which in
many ways is most admirable and useful.

MERITS AND DEFECTS 279

236. After the development of the theory of
limits by the English mathematicians and by such
Continental writers as D'Alembert and Lacroix, it
would hardly seem necessary even for the sake of
brevity to reintroduce the old-time infinitesimal
which could be "dropped" whenever it was very
small, yet stood in the way. But at all times, and
particularly in the eighteenth and beginning of the
nineteenth centuries, there have been mathematicians
who cared little for the logical foundations of their
science. Fascinated by the ease with which the
calculus enabled them to dispose of difficult prob-
lems in the theory of curves, ordinary mechanics,
and celestial motions, they felt more like poets, and
held the sentiments toward logic that a distinguished
bard entertained toward pure intellectualism when
he contemplated the beauties of the rainbow :

" Triumphal arch that fill'st the sky,

When storms prepare to part,
I ask not proud philosophy
To teach me what thou art."

Defects

237. All the eighteenth-century expositions of
the foundations of the calculus — even the British —
are defective. Without attempting an historical
treatment or a logical exposition of later develop-
* ments, we desire to point out briefly what some of
these defects were.

In the first place, the doctrine of fluxions was so
closely associated with geometry, to the neglect of

280 LIMITS AND FLUXIONS

analysis, that, apparently, certain British writers
held the view that fluxions were a branch of geo-
metry. In the preface to the Gentleman's Diary of
London, the new editor, Mr Wildbore, said at the
commencement of his editorship in 1781, "the
doctrine of fluxions depends on principles purely
geometrical, as is very satisfactorily demonstrated
by that incomparable geometer, the late Dr Robert
Simson of Glasgow in his Opera posthuma."

In the second place, as pointed out by Landen
and Woodhouse, there was an unnaturalness in
founding the calculus upon the notions of motion
and velocity. In a real way, these notions seem
to apply only to a limited field in the applications of
the calculus, namely, to dynamics. In other fields,
motion and velocity are wholly foreign concepts
which, if applicable at all, are so only in a figurative
sense.

238. Newtonian writers lay great stress upon
such conceptions as a line generated by the motion
of a point, a surface generated by the motion of a
line, and a solid generated by the motion of a
surface. We have already referred to the pedago-
gical advantages of this view, in teaching beginners.
But as a final logical foundation this view is inade-
quate. Not all continuous curves can be conceived
as traceable by the motion of a point. An example
frequently quoted, in discussions of this sort, is
the curve

sin for xQ,

MERITS AND DEFECTS 281

Let us try to trace this curve by the motion of a
point starting from the origin of co-ordinates. In
which direction must the point move from the
origin ? To answer this question we differentiate,
and find dy / dx=s\n (i / x) — (i / x) cos (i / x). At
the origin we have x = o and y — o. No value can be
assigned to dy / dx, because i / x has no meaning
when;r = o ; moreover, the equation y = x sin (i / x) is
expressly stated above to apply only when x is not
zero. There is, therefore, no way of ascertaining
the direction in which the point must depart from
the origin. Perhaps we can do better if the moving
£>oint is started at another part of the curve. An
attempt to plot the curve reveals the fact that it
lies between two right lines, of which one makes
with the ^r-axis an angle of 45°, the other an angle
of —45°. As the point moves along the curve
toward the origin, the curve is found to oscillate
with ever-increasing rapidity. When we try to
determine the direction by which it jumps into the
origin, we encounter the same difficulty as before.
As long as x is finite, the direction of motion is
determinable. But as soon as we try x=o, the
determination is impossible. This conclusion must
be accepted, in spite of the fact that the curve is
continuous in all its parts, including the origin.
This example illustrates the inadequacy of motion
as a fundamental concept.

239. Difficulties are encountered in the notion of
velocity. Is variable velocity an objective reality ?
Take a body falling from rest. We say that its

282 LIMITS AND FLUXIONS

velocity is ds / dt=gt. At the end of the first
second, the velocity is g. If we ask ourselves, How
far does the body move with the velocity gt we
must admit that no distance can be assigned. We
cannot say that the body moves from a certain point
to the point immediately beneath ; there is no such
point immediately beneath. For, as soon as we try
to locate such a point, it occurs to us that we can
imagine at least one point located between the two
points under consideration. This intermediate point
serves our purposes no better, for a fourth point
located between it and the initial point is easily
detected, and so on, without end. Thus it is seen
that no distance, however small, can be assigned
through which a body falls with a given velocity.
We are thus compelled to reject variable velocity
as a physical fact. What, then, is ds / dt=gtt
Clearly it is merely a limit, a mathematical concept,
useful in mathematical analysis, but without physical
reality. To say that ds / dt represents the distance
a body would fall in unit time after the instant
indicated by /, is to assign it merely hypothetical
meaning, destitute of concreteness. While these
'considerations in themselves may not debar the use
of velocity as a mathematical concept upon which to
build the calculus, they show that the concept is not
as simple as it would seem to be at first approach.

The reader will have observed that in all discus-
sion of limits during the eighteenth century the
question of the existence of a limit of a convergent
sequence was never raised ; no proof was ever given

MERITS AND DEFECTS 283

that a limit actually exists. In this respect the
treatment was purely intuitive.

240. Another defect in the British exposition of
fluxions was in the use of the word " quantity."
No definition of it was given, yet quantities were
added, subtracted, multiplied, and divided. It is
possible to treat quantities or magnitudes without the
use of number. The fifth and tenth books of Euclid's
Elements contain such treatment. We may speak of
the ratio of one magnitude to another magnitude, or
we may speak of the ratio of one number to another
number. Which was meant in the treatment of
fluxions ? Straight lines were drawn and the ratios
of parts of these lines were written down. What
were these the ratios of? Were they the ratios
of the line-segments themselves, or the ratios of
the numbers measuring the lengths of these line-
segments ? No explicit answer to this was given.
Our understanding of authors like Maclaurin, Rowe,
and others is that in initial discussions such phrases
as "fluxions of curvilineal figures," "fluxion of a
rectangle, " are used in a non-arithmetical sense ;
the idea is purely geometrical. When later the
finding of the fluxions of terms in the equations of
curves is taken up, the arithmetical or algebraical
conception is predominant. Rarely does a writer
speak of the difference between the two. Perhaps

" His notions fitted things so well,
That which was which he could not tell."

241. Analytical geometry practically identified
geometry with arithmetic. It was tacitly assumed

284 LIMITS AND FLUXIONS

that to every distance corresponds a number and to
every number there corresponds a distance. Number
was thus given a geometrical basis. This situation
continued into the nineteenth century. This metrical
view involved the entire theory of measurement,
which assumed greater difficulties with the advent
of the non-Euclidean geometries. The geometrical
theory of number became less and less satisfactory
as a logical foundation. Hence the attempts to
construct purely arithmetical theories.1

A good share of those difficulties arose from
irrational numbers, which could not be avoided in
analytical geometry. This occurrence is not merely
occasional ; irrational ratios are at least as frequent
as rational ones. What is an irrational number ?
How do we operate with irrational numbers ? What
constitutes the sum, difference, product or quotient,
when irrational numbers are involved ? No explicit
answer was given to these questions. It was tacitly
assumed without fear, that it is safe to operate with
irrational numbers as if they were rational. But
such assumptions are dangerous. They might lead
to absurdities. Even if they do not, this matter
demands attention when mathematical rigour is
the aim.

242. Perhaps it may be worth while to recall to
the reader's mind illustrations of the danger result-
ing from taking operations known to yield consistent

1 For a historical account of the number concept and the founding
of the theory of transfinite numbers during the nineteenth century,
read Philip E. B. Jourdain's "Introduction" to Cantor's Transfinite
Numbers, The Open Court Publishing Co., 1915.

MERITS AND DEFECTS 285

results when a certain limited class of numbers is
involved, and applying them to numbers of a more
general class. Suppose a and b to be rational,
positive numbers, not zero ; we find, let us agree,
consistent results in the operation a-\-b^ a — b when
a>b, and #x£, and a-^-b. Let us now consider
the class composed of rational numbers, both
positive and negative ; suppose, moreover, that
we introduce o in order to give interpretation to
the operation a — a. If in this extended class of
numbers we admit the four operations a-\-b, a — b,
axb, a-±b, trouble arises even after due considera-
tion has been given to the negative numbers. There
may arise the following well-known paradox. Let
# = £=!, then cP — fi^a — b. Divide both sides of
the last equation by a — b, and we have a + b= I, or
2=1. Where is the difficulty? The answer is
known to every schoolboy : We have used a— b, or
o, as a divisor ; we have extended the operation of
division to the larger class of numbers, and to
zero, without first assuring ourselves that such an
extension is possible in every case ; division by zero
is inadmissible.

243. A less familiar example is the following.
Let us suppose that, for real exponents, it is estab-
lished that (A.x)y = A-**. When we apply this process
to imaginary exponents, trouble arises. Take the
equation ^aww/ = ^aw", where m and n are distinct
integers, i= J~i> TT= 3'I4I59 • • •> and 2=2718
. . . That this equation holds is evident, for e*"***
= cos 2mir + / sin 2;;/?r = cos 2mr + / sin 2n^ = e2™. I f

286 LIMITS AND FLUXIONS

both sides of e*miri = e* wi are raised to the power i / 2,
we obtain e~m* = e~H*. Here all the letters stand for
real numbers ; since m and n are not equal to each
other, this last equation is an absurdity. The
assumption that a rule of operation valid for real
exponents was valid also for imaginary exponents,
has led to papable error.

Examples of this sort emphasise the need of
caution when operations, known to be valid for a
certain class of numbers, are applied to numbers
belonging to a larger class. Special examination
is necessary. These remarks are pertinent when
operations applicable to rational numbers are ex-
tended to a class which embraces both rational and
irrational numbers. What are the numbers called
irrational ? It is hardly sufficient to say that an
irrational number is one which cannot be expressed
as the ratio of two rational numbers. A negative
definition of this sort does not even establish
the existence of irrational numbers. Considerable
attention has been paid to the definition of irrational
numbers as limits of certain sequences of rational
numbers. Thus, J '2 may be looked upon as the
limit of the sequence of rational fractions obtained
by the ordinary process of root-extraction, namely,
the sequence, I, 1*4, 1*41, 1*414, 1*4142, . . .
This attempt to establish a logical foundation
for irrational numbers was not successful. We
endeavour, in what follows, to make this matter
plainer.

244. Let us agree that in building up an arith-

MERITS AND DEFECTS 287

metical theory we have reached a development of
rational numbers (integers and rational fractions).
We wish, next, to define limit and also irrational
number. An early nineteenth-century definition of
limit was : ''When the successive values attributed
to a variable approach a fixed value indefinitely so
as to end by differing from it as little as is wished,
this fixed value is called the limit of all the others."
Since, according to our supposition, we are still in
the field of rational numbers, this limit, unless it
happens to involve only rational* numbers and to be
itself only a rational number, is, in our case, non-
existent and fictitious.

If now, as stated above, an irrational number is
defined as the limit of certain sequences of rational
fractions, trouble arises. The existence of such a
limit is often far from evident. But aside from that
general consideration, the difficulty of the situation
in our case is apparent : Irrational numbers are
limits, but limits themselves are non-existent or
fictitious, unless they are rational numbers. To
avoid this breakdown in the logical development,
it was found desirable to define irrational number
without using limits.

245. With the view of avoiding the use of limits
in the definition of irrational number, and at the
same time avoid inelegant and difficult assumptions,
involving complicated considerations relating to the
nature of space,1 devices were invented by several

1 On this point consult the article " Geometry " in the Encyclopedia
Britannica, nth edition, the part on Congruence and Measurement.

288 LIMITS AND FLUXIONS

logicians independently, which freed the number con-
cept from magnitude and established number theory
on the concept of order. Chief among the workers
in this field were Meray, Weierstrass, Dedekind,
and Georg Cantor. It is to them that we owe re-
presentations of number, both rational and irrational,
which have yielded a much more satisfactory theory
of limits, and in that way have vastly improved the
logical exposition of the differential calculus. These
theories have brought about the last stages of what
is called the arithmetisation of mathematics. As
now developed in books which aim at extreme
rigour, the notion of a limit makes no reference to
quantity and is a purely ordinal notion. Of this
mode of treatment the eighteenth century had never
dreamed.

ADDENDA TO §§ 54, 58, 73

246. ADDITIONAL data on the fundamental concep-
tions relating to fluxions and on the use of the
Newtonian and the Leibnizian notations in England
during the lifetime of Newton are contained in
George Cheyne's Philosophical Principles of Religion,
Part II, London, 1716. Part I of this book
appeared first in 1705. Like Berkeley's Analyst,
which was written later, Cheyne's book, Part I and
Part II, had for its primary purpose the refutation
of atheism. Cheyne says in his preface to the third
edition of Part I, "that Atheism, may be eternally
confounded, by the most distant Approaches to the
true Causes of natural Appearances. And that if
the Modern Philosophy demonstrates nothing else,
yet it infallibly proves Atheism to be the most
gross Ignorance."

Part 1 1 of Cheyne's book consists of three chapters
and of seven pages of " Additions." He says in his
preface to this part that, excepting one short note,
the third chapter and the "Additions" are "what
the reverend and ingenious Mr. John Craig sent me
about seven years ago, when 1 desired him ... to
write me down his Thoughts on, correct or alter,

289 19

290 LIMITS AND FLUXIONS

what I had formerly published on this Head in the
first Edition of this Work."

Cheyne uses in Part II, p. 20, the notation x to
denote a distance B£ when he supposes ' ' b infinitely
near to B." In § 58 we pointed out that in 1704
Cheyne wrote once x= i, but nowhere in the present
book does x denote a finite quantity. He argues
that I /,r=oo i, that I / 0=00 ; hence that;r=0, or
"relative nothing," which is " the least Part of the
Finite, to which it is related or compared." On
p. 21 he calls x "an infinitely little Part of #." On
p. 12 he speaks of the "absolute infinite" as " ad-
mitting of neither Increase, nor Diminution, or of
any Operation that mathematical Quantity is sub-
jected to," while (p. 13) "absolute nothing" is
"neither capable of increasing nor diminishing, nor
of any wise altering any Mathematical Quantity to
which it is apply'd, but stands in full opposition to
absolute Infinite." On the other hand, "indefinite"
or ' ' relative infinite " quantities (p. 29) ' ' are not
properly either Finite or Infinite, but between both."
The " relative nothing " (p. 8) "is an infinitely little
Quantity, as it stands related to a given Finite,
by the perpetual Subtraction of which from it self
it is generated. Let o stand for relative nothing.
Thus 01 is a relative infinitely little Quantity, as it
stands related to Unity, by the perpetual Subtrac-
tion of which from it self, it is generated ; that is
0i = i — i + i — i + i — i + i — i ec., and oa is an
infinitely little Quantity, as it stands related to
the given Finite a, by the perpetual Subtraction

ADDENDA 291

of which from it self, it is generated ; that is
oa — a — a + a — a + a — a ec." In the " Advertize-
ment" following p. 190 this is further explained
thus : * ' Relative Nothing is said here to be generated
by a perpetual Subtraction, tho' the Signs by alter-
nately + and — . For these Reasons, because
relative Infinite, was said to be generated by a
perpetual Addition, and because that after the first
Term, every two succeeding ones in relative Nothing
I is equivalent to 01 thus I — i-fi — i + i — I, &c.
. . . = I — 01 — '01 — 01 &c."

247. These explanations are intended by Cheyne
merely as introductions to the later chapters, par-
ticularly that by John Craig, who (p. 167) declares
that o cannot be an absolute nothing, ' ' for an
infinite Number of absolute Nothings cannot make I,
but by o is understood an infinitely small part, as
in the calc. diff. dx is an infinitely small part of lar,
so that dx is as o to x : Not that dx is absolutely
nothing, for it is divisible into an infinite Number of
Parts, each of which is ddx. " To make the point
still plainer, John Craig continues (p. 168) : "But
then it may be inquir'd what is the Quotient that
arises from the Division of I by absolute Nothing.
I say there is no Quotient because there is no
Division : Therefore it is a Mistake to say the
Quotient is I or Unity undivided, which is demon-
strably false, neither is the Quotient = o. For
properly speaking there is no Quotient, and there-
fore it is an Error to assign any. In like manner,
it is an Error to say, that o x a makes the Product

292 LIMITS AND FLUXIONS

o ; for properly speaking there is no Product. It is
true, this of Multiplication has no influence upon
Practice, but that of Division has. From hence it
appears, that a Curve is said to meet with its Asymp-
tote, when the Ordinate is infinitely little." Then
follows a startling view which had been held about
sixty years before by John Wallis in his Arithmetica
Infinitorum, I655,1 but Craig makes no reference to
him. Craig argues (p. 169): "This same Notion
does explain how it comes to pass that I divided by
a negative Number gives a Quotie'nt greater than
Infinite." Curiously, he represents the logarithmic
curve j/ = log;r as crossing the j/-axis at y— — oo ,
for since the curve approaches the y - axis in-
finitely near when positive x approaches zero, "we
may conceive the Logarithmic Curve continued as
intersecting " the j/-axis, so as to form * ' one con-
tinued Curve." Accordingly negative numbers have
logarithms that are real and negative. His further
argument amounts to this : For values of x that are
equal to I divided by a negative number, y\\\.y = log^r
is negative and is less than its value — oo arising
when x—o (presumably in the sense that — 2 < — i).
"Ergo;tr is a Number greater than infinite." Con-
sidering the approach of the logarithmic curve
towards its asymptote, Craig says (p. 170) that
"here it is observable, that there are affirmative
Numbers less than nothing denoted by the several
Powers vfdx, as dxz, dx*, ec. , or by the second, third,
ec. Differences, and these Numbers may be aptly

1 Wallis, Opera, I, p. 409, Prop. CIV.

ADDENDA 293

represented by the Ordinates of the logarithmic
Curve," continued from y=— oo away from the
origin when dxn is affirmative, or towards the origin
when dx» is negative. In the "Additions," p. 185,
Craig devotes six pages to "An Answer to Mr.
Varignon's Reflections upon Spaces greater than
infinite," in which Craig uses the Leibnizian symbol

/five times, as in/: x~edx = — — . ' Nowhere in the

book under consideration does Craig use the nota-
tion of Newton. The "Additions" are dated
"September 23d, 1713."

248. George Cheyne was a pupil of the Scotch
physician, Archibald Pitcairne (1652-1713), who is
the author of two books on fluxions (which we have
not seen), viz. Fluxionum Methodus inversa ; Sive
Quantitatum Fluentium Leges generaliores. Ad
celeberrimum virum, Archibaldum Pitcarnium,
MedicumEdinburgensem\ and RudimentorumMethodi
Fluxionum inverses Specimina adversus Abr. de
Moivre. Pitcairne's mathematical bent more or
less influenced his medical theories. He liked to
ridicule others, and was himself ridiculed in a publica-
tion, Apollo Mathematicus ; or, the Art of curing
Diseases by the Mathematicks, according to the
Principles of Dr. Pitcairne, 1695.

INDEX

Absolute motion, 86, 179.
nothing, 219, 291.
space, 178, 179.
"Achilles," 97, 125.
Agnesi, M. G., 247-250.
Alembert, d', see D'Alembert,
Analyst (Berkeley's), 34, 57-95,
101, 117, 128, 141, 142, 145,
148, 149, 151, 153, 155, 157,

158, 165, 179, 181, 183, 189, |
199, 229, 251, 253, 254, 263,
266, 289.

Analytical Society at Cambridge,

274.

Angle of contact, 42.
Antecedental calculus, 235-239.
Apollonius, 31, 155.
Archimedes, 181, 187, 253.
Aristotle, 166, 256.
Arithmetisation, 225-239, 288.

Babbage, Charles, 271, 274.

Barrow, Isaac, I, 48, 52, 65, 132,
185 ; his definition of velocity,
185.

Kayes, Thomas, 157.

Berkeley, George (Bishop), 2, 34,
56-95, 87-91, 94-96, 101, 112,
123, 148, 149, 151, 158, 163,
168, 171, 178, 183, 184, 188,
190, 199, 200, 203, 205, 216,

2l8, 222, 229, 250, 263, 265,

266 ; his first reply to Jurin
and Walton, 72-78 ; his second i
reply to Walton, 85-87 ; Berke- |
ley's lemma, 59, 71, 84, 93,

159, 160, 161, 168, 200, 265 ;
Berkeley ten years after, 178,
179.

Bernoulli, U., 41, 274, 278.
John, 31, 32, 44, 114, 140, 274,

278.
Binomial theorem, 60, 233, 237,

247, 264.
Bossut, Ch., 269.
Bougainville, L. A. de, 204.
Brewster, David, 26.
Buffon, Comte de, 203-205.
Buonaparte, 90.

Calculus, pre-history of, I ; contro-
versy on invention of, 38, 39,
47, 49 ; see Differential calculus.

Cantabrigiensis, 209, 210, 215.

Cantor, Georg, 284, 288.
Moritz, 235.

Carnot, L. N. M., 62, 262, 263.

Carre, L., 41.

Cause and effect, 79, 166, 174, 185,
192-194, 210, 212, 257.

Cavalieri, i, 26, 100, 116, 132.

Cheyne, George, 40, 41, 289-290,

293-

Clairaut, A. C, 274, 278.
Clarke, John, 47.
Clerk, G. L. Le, see Buffon, Comte

de.

Collins, John, 26, 47, 48, 112, 244.
Colour device in numbers, 137.
Colson, John, 149-154, 164, 165,

182, 247, 248.
Commercium EpistolicTtm, 26-29,

36, 47-49, 52, 55, 112, 204,

244.
Compensation of errors, 62, 63, ,

262.

Cotes, Roger, 31, 39, 55, 211.
Cournot, A. A., 94.

294

INDEX

295

Cousin, J. A. J., 269, 270.
Craig, John, 37-39, 55, 274, 289,
291-293.

Daily Gazetteer, 212.

D'Alembert, 204, 238, 239, 254,

255, 257, 263, 265, 268, 271-

273, 275, 278, 279.
Davis, William, 253.
Dealtry, William, 252.
Dedekind, R. , 288.
De 1' Hospital, 41, 50, 51, 53, 54,

66, 84, 169, 248.
De Moivre, A., 39-41, 243.
De Morgan, A., 26, 29, 32, 34, 38,

39, 90, 91, 94, 238, 239, 274.
" Dichotomy," 119.
Diderot, D., 275.
Differential calculus, 45, 48, 52, 58,

6l, 62, 114, Il8, 139, 156, 222,

225, 239, 249, 258, 260, 261,
271, 277, 288.

method of Leibniz, 28, 29, 37, 38,

49, 2\7',see Differential calculus.

Ditton, H., 40, 41, 44, 46, 47, 50,

53, 55^ 56-

Duhamel, J. M. C., 94.
Duillier, Fatio de, 39, 55.

Edinburgh Review, 249.
Edleston, J., 31, 32, 40.
Effect, see Cause and effect.
Emerson, William, 178, 192-194,
206, 209, 213, 218, 221, 223,

243. 259.
Encyclopedia Britannica, 240, 269,

273, 287.
Enestrom, G., 30.
Errors, not to be neglected, 19, 23,

28, 32, 34 ; see Infinitesimals,

Infinitely little quantities.
Euclid, 7, 10, 31, 119, 155, 184,

187, 191, 222, 253.
Euler, L., 139, 140, 258. 265, 274.
Evanescent quantities, 6, 8-10, 19,

20, 23, 24, 26, 35, 62, 63, 68,

70, 71, 73. 76, 77, 79, 81,105,

106, 116, 131, 135, 137, 156,
162, 165-168, 191, 247, 255.

Exhaustion, method of, I, 76, 100,

107, 128, 187, 272, 273.

Fatio de Duillier, 39, 55.

Fenn, John, 274.

Flowing quantities ; see Fluents.

Fluents, 15, 16, 18, 21, 22, 25, 27-
29, 43, 49, 55, 7°, 89," 112,
113, 159, 162, 173, 174, 192,
J95, J99, 2OI> 202, 209, 211,
213-215, 246, 273.

Fluxions, criticisms of, 58, 72, 90,
225, 229, 232, 235, 236, 255-
276 ; definition of — by Emerson,
192, in Encyclopedia Britan-
nica, 240, by Maclaurin, 184,
by I. Newton, n, 13-16, 18,
21, 22, by Simpson, 169, 179,
195, 21 1, 213, 219, 244;
erroneously defined, 39-56, 71,
156, 171, 172, 198, 240, 248 ;
notation of, 15, 1 6, 28-33, 38-
40, 43, 46, 48, 49, 54, 186,
202, 248, 270, 273, 289-293 ;
pre-history of, I ; second, third,
etc., fluxions, 39, 42, 47, 55,
56, 63, 74, 77, 80, 82, 89, 90,
173, 182, 196, 199, 241, 272.

Eraser, A. C., 178, 188.

Function, 235, 269.

Galileo, I.

Gentleman 's Diary, 280.

"Ghosts of departed quantities,"

63, 68, 73, 85, 216, 250.
Gibson, G. A., 93, 96.
Glenie, James, 235-239.
Graves, J. T., 276.

R. P., 38, 90, 91, 94, 180, 274,

276.

Gray, G. J., 165.
Gregory, David, 39.

James, 48.

St. Vincent, I, 99.

Hales, William, 267-269, 275.

Halley, E., 39, 57.

Hamilton, W. R., 38, 90, 91, 94,

1 80, 274, 276.
Harris, John, 40, 41, 55.
Hayes, Ch., 41-43, 53, 55, 56.
Heath, Robert, 207. 210, 212, 213,

229.
Hellins, John, 247-250.

296

LIMITS AND FLUXIONS

Herschel, J. F. W., 271, 274.
Hobbes, Thomas, 163.
Hodgson, James, 155-157.
Holliday, F., 243, 244.
Hospital, De 1', 41, 50, 51, 53, 54,

66, 84, 169, 248.
Hutton, Charles, 210, 244, 245.

Imaginary quantities, 154.

Imagination, straining of, 120—123,
125, 147, 154, 195, 214, 217.

Incommensurables. 7, 10.

Indivisibles, method of, 5, 7, 8,
24, 29, 100, 106, 114, 116,
117, 127, 128, 132, 133, 139,

150, 152, 154, 183, 204, 205, ,

243-
Infinite divisibility, 5, 7, 10, 61, 89,

Infinite series, method of, 20, 24,

164, 221.
Infinitely little quantities, 15-17,

20, 21, 24. 26-28, 30-36, 39,

40-43, 45, 47, 48, 50, 51, 53, ;
54, 56, 58, 61, 76-78, 80, 86, !
94, 114, 117, 118, 123, 132,
133, J37, H7, 150-156, 162,
168, 169, 174, 182, 183, 196,
!97, 199-202, 220, 243, 248, ;
258, 261, 277, 290-291 ; avoid-
ing their use. 142, 162, 169, '
171, 183, 184, 186, 189, 275 ; '
see Infinitesimals.

Infinitesimals, 29, 32, 35, 36, 46,
50, 56, 58, 59, 72, 74-76, 83,
89, 94, in, 114, 118, 147, 182, ;
184, 201, 206. 207, 241, 250,
263, 275, 277-279.

Infinity, 162, 221, 249; absolute, \

151. 163, 292; relative, 151,
162, 163, 183, 184, 290.

Irrational numbers, 284, 286-288.

Jack, Richard, 222, 223.

Jones, William, 17, 43, 243.

Jourdain, Philip E. B.. 29, 263,
274, 284.

Jurin, James, 57, 64, 72, 91, 93,
139, 140. 189, 190, 199, 203,
205, 206, 254, 272 ; Jurin's
first reply to Berkeley, 64—69 ;

second reply, 80-85 '•> contro-
versy with Robins, 96-148, 179;
controversy with Pemberton,
129-138 ; explains history of
controversy with Robins, 145-
146.
Juvenal, 168.

Keill, John, 31, 48.
Kelland, Philip, 189.
Kirkby, John, 225-230.
Klopstock, 1 88.
Kowalewski, G., 17.

Lacroix, S. F., 255, 260-262, 268-

275, 279-
Ladies Diary, 207, 209, 210, 212,

219, 221, 222, 229.

Lady* s Philosopher, 219.

Lagrange, J. L., 94, 181, 239, 255,
259, 266, 270-272, 274, 278 ;
his Fonctions analytiques, 255,
259, 263, 266, 272.

Landen, J., 215, 222, 231-235, 239,
257, 258, 261, 262, 266, 270,
273, 280.

Laplace, P. S,, 268, 278.

Last ratio ; see Ultimate ratio.

Lee, Sidney, 181, 263.

Legendre, A. M., 268, 278.

Leibniz, G. W., I, 28, 32, 33, 37,
38, 41, 47-49, S2, 53, 55, 94,
106, 112-114, 118, 175, 176,
199, 248, 254, 262, 274; Differ-
ential method of, see Differen-
tial calculus.

Leibnizian notation, 32, 37, 38, 48,
49, 53, 54, 139, 254, 274, 278,
289, 291-293.

Leslie, John, 273.

L'Huilier (Lhuilier), S. A. T., 269.

Limit, i, 3-5, 7-10, 25, 33; 35, 36,
62, 70, 75-78, 83, 89, 92, 93,
97,99, 101, 104, 1 06, 116, 119,
129, 146, 147, 183, 185, 189,

191, 2l8, 220, 238, 241, 242,
247, 250, 251, 254, 255, 258,
259, 263, 265-269, 271, 272,
275, 276, 279, 282, 286-288 ;
limit not reached, 97, 102, 107,
116, 119, 131, 147, 186, 189,

INDEX

297

221, 241, 242, 275; limit is
reached, 97, 98, 101, 103, 119.
123, 124, 147, 157, 190, 271.

Locke, John, 69.

Lyons, Israel, 201.

Maclaurin, Colin, 181, 199, 203-
205, 250, 253, 254, 256, 263,
266, 268, 272, 273, 277, 283 ;
on infinitely small magnitudes,
182; life of, 183.

Maclaurin's Treatise of Fhixions,
149, 175, 181-189, 223, 254,
263 ; French translation, 189.

Martin, Benjamin, 171, 172, 178,
179.

Maseres, Baron, 248, 267, 268.

Mathematical Exercises (Turner's),

2IO, 2l8, 219.

Mathematician, 210.

" Maximinority," 188.

Meray, H. C. R., 288.

" Minimajority," 188.

Minute Mathematician, 80, 136,
138.

Moivre, De, 39.

Moments, 11-14, !6, 25-28, 30, 31,
33, 39, 40, 42, 46, 47, 49, 58,
59, 66, 67, 70, 71, 73, 74, 76-
78, 89, 91, 92, 99, 100, 105,
106, 116, 118, 123, 128, 147,
151-155, 159, 161, 172, 176,
183, 192, 193, 196, 197, 201 ;
notation for, 25, 27, 28, 31;
origin of term, 26.

Moment of a rectangle, 12, 13, 58,
66, 71, 74, 75,79,82, 83, 91,
92, 151-

Monthly Review * 200, 209, 210,

215, 222, 224, 234, 255, 259-

262, 267, 268.

Morgan, De ; see De Morgan, A.
Morgan, James, 210.
Motte, A., translator of Principia,

4, 12, 103.
Muller, John, 162-164.

Napier, J., i.

Nascent magnitudes, II, 13, 22, 43,
44, 58, 59, 62, 66, 70, 73, 74,
77, 81, 84, 106, 113, 116, 118,

126, 135, 156, 161, 162, 165-
167, 172, 177, 191, 194, 243,
245.

Navier, L. M. H., 94.

Newton, Isaac, 2-36, 38-41, 43,
49, 52, 53, 55, 56, 61, 64, 65,
67~7o, 75, /8, 81-86, 88, 91,

92, 96, IOO, IO2, IO4, III-II4,
Il8, 122, 123, 127 129, 130,

!33, 137, Hi-143' H7, 152,
J54, 155, J57, 158, 160, 161,
164, 165, 168, 170-173, 175,
181-183, 191, 192, 195, 198,

2O3-2O5, 211, 214, 219, 22O,
229, 235, 241, 244, 248, 252,
254, 256, 257, 26l, 264, 268,
2/0, 272.

Newton's Analysis per cequationes t

17, 36, 48, 190, 204.
lemmas, 3, 43, 75-77, 83, 91,

101-103, 105, 107, 109, no,

115, 118, 124, 128. 131, 132,

J34, 135, 137, 138, 145, 146,

151, 158, 189, 190, 203, 241,

242, 246, 250, 254.
manuscripts, 29.
Method of Fluxions, 149, 164,

248.
notation for fluxions; see Fluxions,

notation of.
Opticks, 17, 127, 204.
Principia, quotations from, 2-14,

43 ; see Principia (Newton's).
Quadratiira Curvaruin ; see Quad-

ratura Curvarum.
use of infinitely little quantities,

29, 30, 32, 33-36, 155-
Newton, T., 250, 251.
Nieuwentiit, B., 41, 45, 46, 102.
Notation for fluxions, see Fluxions,

notation of ; for calculus, see

Leibnizian notation.
Nothing, absolute, 219, 290-291 ;

relative, 290-291.

Oldenburgh, 26.
Ostwald's Klassiker, 17.

Palladium, 210, 212, 219.
Parent, A., 114.
Pascal, B., 132.

298

LIMITS AND FLUXIONS

Peacock, George, 271, 272, 274.

Pemberton, H., 96, 125, 129-138,
147, 165.

Petvin, John, 230.

Pezenas, Esprit, 189.

" Philalethes Cantabrigiensis," 64,
72, 82, 101, 106, 108-110, 117,
122, 124-128, 133, 136-138,
140-144, 181, 199, 204, 205 ;
see Jurin, James.

" Phileleutherus Oxoniensis," 133.

Philosophical, Transactions. 26, 38-
41, 112, 118, 157, 185, 235,
244, 274.

Pitcairne, Archibald, 293.

Poisson, S. D., 94.

Portsmouth collection, 30.

Present State of the Republick of
Letters, 99, 101, 106, 109, in,
117, 123-125, 129, 130, 133,
135, 142, 143, 154, 162.

Price, B., 278.

Prime and ultimate ratios, 5, 6, 8,
19, 20, 22, 24, 29, 33, 35, 36,
62, 71, 80, Si, 86, 88, 89, 96,
100, 109, 113, 114, 117, 118,
122, 124, 126, 128, 130, 141,
143, 147, 166, 186, 190, 241,
243, 250, 258, 265, 267, 273.

Principia (Newton's), 28, 30, 31-
33, 36, 53, 67-69, 74-77, 86,
101-104, 107, no, 126, 128,
I29, T35> J58, 190, 219, 241-
243, 253 5 quotations from, 2-
H, 43-

Quadrature/, Curvarum (Newton's),
17-26, 28, 31-34, 36, 43, 50,
53, 67, 69, 71, 74, 76, 77, 112,
124, 130, 135-137, 156, 179,
190, 246.

Quadrature of curves ; see Quadra-
tura Curvarum.

Raphson, Joseph, 49, 55

Ratios, first and last ; see Prime

and ultimate ratios.
Relative infinite ; see Infinity,

relative.
Relative motion, 86.

Relative nothing ; see Nothing,
relative.

Republick of Letters', see Present
State of the Republick of Letters.

Residual analysis (Landen's), 232-
235, 239, 258, 273.

Rigaud, S. P., 14, 29, 30.

Robartes, F., 41.

Roberval, I.

Robins, B., 96-148, 179, 181, 188,
189, 199, 242, 250, 254, 266,
267, 272-275 ; Robins's Mathe-
matical Tracts, 202-206 ;
Robins's Discottrse, 96-100,
115, 141, 142, 267, 268, 275;
explains history of controversy
with Jurin, 141-145.

Rowe, John, 172, 173, 175, 178,
179, 195, 202, 253, 283.

Rowning, John, 198-200.

Saunderson, Nicholas, 150, 197,

198, 204, 242.

Simpson, Thomas, 169-171, 175,

178-180, 194-196, 206, 209,

210-215, 2l8, 220, 221, 223,

224, 244, 253, 259, 277 ; his
definition of fluxions. 244.

Simson, Robert, 280.

Sloane, H., 48.

Smith, James, 165-169.

Smith, Robert, 139. 140, 201.

Solidus, use of, 12.

Stevin, S., 99.

Stewart, John, 17, 21, 25, 190-192.

Stirling, James, 50, 55.

Stockier, G., 259.

Stone, E., 50, 51, 53-56, 169, 248,

259.

St. Vincent, Gregory, i, 99.
Stubbs, J., 177.

Tacquet, Andrew, 99.

Tannery, Jules, 137.

Taylor, Brook, 50.

Thorp, R., 241-243 ; translator of

Principia, 3, 8.
Torelli, G., 257.
Truth Triumphant, 212, 213, 215,

218, 222, 223.
Turner, John, 210-214, 218, 219.

INDEX

299

Ultimate ratios, 6, 7, 9, 10, 19, 22-
24, 35, 68, 70, 78, 86, 98, 99,
103. IJ5> !53> iS8-1^!, 168,
191, 219, 223, 242.

Ultimate velocity, 7, 10 ; see
Velocity.

Ultimators, doctrine of. 225.

Valerius, Lucas, 99.

Varignon, P., 293.

Velocity, n, 13-18, 21, 28-30, 32,
33. 36, 43-45, 49, 5°» 58, 62,
63, 66, 70, 71, 74, 76-80, 83,
85, 87, 88, 93, 106, 112, 123,
151, 152, 158, 163, 164, 166,
169-171, 173, 175, 184, 185,
193-197, 199, 200, 201, 207,

2O9, 212, 214, 215, 217, 219,
220, 231, 235, 244, 246, 252,

261, 263, 270, 277, 280 ; velo-
city, ultimate, 6, 9, 36 ; velo-
city criticised as a fundamental
concept, 255, 264, 280-282.

Vince, S., 245-247, 277.

Vivanti, G., 235.

Wallis, John, 41, 100, 292 ; Wallis's
Algebra, 14, 30, 32, 33, 39.

Walton, John, 57, 69, 77, 85, 86,
91-96, 148, 185, 199; his first
reply to Berkeley, 69-72 ; his
second reply, 78-80 ; second
edition of second reply, 87.

Warburton, Bishop William, 73.

Waring, E. , 259.

Weierstrass, K., 288.

Weissenborn, H. , 92.

West, William, 202.

Whately, 90.

Whiston, William, 197.

Wildbore, 280.

Wilson, James, 96, 99, 189, 202-
206.

Witting, A., 30.

Wood, James, 245.

Woodhouse, R., 93, 263-267, 274,
280.

Works of the Learned, 135-137,
171.

Zeno (of Elea), 90, 119, 125, 217.

PRINTED IN GREAT BRITAIN BY NEILL AND CO., LTD., EDINBURGH.

QA Cajori, Florian
303 A history of the concep-
C25 tions of limits and fluxions
in Great Britain

physical &
^Applied ScL

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